GCSE Maths: Standard form: large and small numbers (Higher)

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Check that the front number is at least 1 and less than 10.

1. Remember the rule. A number is in standard form if it is written as A × 10ⁿ, where A is at least 1 and less than 10, and n is a whole number.
   1 ≤ A < 10 and n a whole number
2. Check the front number A. A is the number in front of the × 10.
   A = 3.4: this is at least 1 and less than 10 ✓
3. Check the power of 10. The index must be a whole number (it can be positive, negative or zero).
   n = 5, a whole number ✓
4. Decide.
   Yes, it is in standard form.

Check. 3.4 is inside the range 1 to 10, so the answer is yes. ✓

Watch out. A can equal 1 but it can never be 10. The power of 10 can be 0, positive or negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Check that the front number is at least 1 and less than 10.

1. Remember the rule. A number is in standard form if it is written as A × 10ⁿ, where A is at least 1 and less than 10, and n is a whole number.
   1 ≤ A < 10 and n a whole number
2. Check the front number A. A is the number in front of the × 10.
   A = 34: this is 10 or more ✗
3. Check the power of 10. The index must be a whole number (it can be positive, negative or zero).
   n = 4, a whole number ✓
4. Decide. A must be between 1 and 10. It should be 3.4 × 10⁵.
   No, it is not in standard form.

Check. 34 is outside the range 1 to 10, so the answer is no. ✓

Watch out. A can equal 1 but it can never be 10. The power of 10 can be 0, positive or negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Check that the front number is at least 1 and less than 10.

1. Remember the rule. A number is in standard form if it is written as A × 10ⁿ, where A is at least 1 and less than 10, and n is a whole number.
   1 ≤ A < 10 and n a whole number
2. Check the front number A. A is the number in front of the × 10.
   A = 0.5: this is less than 1 ✗
3. Check the power of 10. The index must be a whole number (it can be positive, negative or zero).
   n = 3, a whole number ✓
4. Decide. A must be between 1 and 10. It should be 5 × 10².
   No, it is not in standard form.

Check. 0.5 is outside the range 1 to 10, so the answer is no. ✓

Watch out. A can equal 1 but it can never be 10. The power of 10 can be 0, positive or negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Check that the front number is at least 1 and less than 10.

1. Remember the rule. A number is in standard form if it is written as A × 10ⁿ, where A is at least 1 and less than 10, and n is a whole number.
   1 ≤ A < 10 and n a whole number
2. Check the front number A. A is the number in front of the × 10.
   A = 9.99: this is at least 1 and less than 10 ✓
3. Check the power of 10. The index must be a whole number (it can be positive, negative or zero).
   n = −2, a whole number ✓
4. Decide.
   Yes, it is in standard form.

Check. 9.99 is inside the range 1 to 10, so the answer is yes. ✓

Watch out. A can equal 1 but it can never be 10. The power of 10 can be 0, positive or negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Check that the front number is at least 1 and less than 10.

1. Remember the rule. A number is in standard form if it is written as A × 10ⁿ, where A is at least 1 and less than 10, and n is a whole number.
   1 ≤ A < 10 and n a whole number
2. Check the front number A. A is the number in front of the × 10.
   A = 10: this is 10 or more ✗
3. Check the power of 10. The index must be a whole number (it can be positive, negative or zero).
   n = 2, a whole number ✓
4. Decide. A must be between 1 and 10. It should be 1 × 10³.
   No, it is not in standard form.

Check. 10 is outside the range 1 to 10, so the answer is no. ✓

Watch out. A can equal 1 but it can never be 10. The power of 10 can be 0, positive or negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Check that the front number is at least 1 and less than 10.

1. Remember the rule. A number is in standard form if it is written as A × 10ⁿ, where A is at least 1 and less than 10, and n is a whole number.
   1 ≤ A < 10 and n a whole number
2. Check the front number A. A is the number in front of the × 10.
   A = 6: this is at least 1 and less than 10 ✓
3. Check the power of 10. The index must be a whole number (it can be positive, negative or zero).
   n = 0, a whole number ✓
4. Decide.
   Yes, it is in standard form.

Check. 6 is inside the range 1 to 10, so the answer is yes. ✓

Watch out. A can equal 1 but it can never be 10. The power of 10 can be 0, positive or negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 6000 is 6.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 6
3. Count how many places the decimal point has moved. Starting from 6000 the decimal point has to move 3 places to the left to sit after the 6. Each place moved is a factor of 10.
   6000 → 6: 3 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 3
5. Write the answer. Put A and the power of 10 together.
   6 × 10³

Check. Convert back: 6 × 10³ = 6000 = 6000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 45,000 is 4.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 4.5
3. Count how many places the decimal point has moved. Starting from 45000 the decimal point has to move 4 places to the left to sit after the 4. Each place moved is a factor of 10.
   45000 → 4.5: 4 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 4
5. Write the answer. Put A and the power of 10 together.
   4.5 × 10⁴

Check. Convert back: 4.5 × 10⁴ = 45000 = 45,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 380,000 is 3.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 3.8
3. Count how many places the decimal point has moved. Starting from 380000 the decimal point has to move 5 places to the left to sit after the 3. Each place moved is a factor of 10.
   380000 → 3.8: 5 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 5
5. Write the answer. Put A and the power of 10 together.
   3.8 × 10⁵

Check. Convert back: 3.8 × 10⁵ = 380000 = 380,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 5,600,000 is 5.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 5.6
3. Count how many places the decimal point has moved. Starting from 5600000 the decimal point has to move 6 places to the left to sit after the 5. Each place moved is a factor of 10.
   5600000 → 5.6: 6 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 6
5. Write the answer. Put A and the power of 10 together.
   5.6 × 10⁶

Check. Convert back: 5.6 × 10⁶ = 5600000 = 5,600,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 91,000,000 is 9.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 9.1
3. Count how many places the decimal point has moved. Starting from 91000000 the decimal point has to move 7 places to the left to sit after the 9. Each place moved is a factor of 10.
   91000000 → 9.1: 7 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 7
5. Write the answer. Put A and the power of 10 together.
   9.1 × 10⁷

Check. Convert back: 9.1 × 10⁷ = 91000000 = 91,000,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 720 is 7.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 7.2
3. Count how many places the decimal point has moved. Starting from 720 the decimal point has to move 2 places to the left to sit after the 7. Each place moved is a factor of 10.
   720 → 7.2: 2 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 2
5. Write the answer. Put A and the power of 10 together.
   7.2 × 10²

Check. Convert back: 7.2 × 10² = 720 = 720. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.7 is 7.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 7
3. Count how many places the decimal point has moved. Starting from 0.7 the decimal point has to move 1 place to the right to sit after the 7. Each place moved is a factor of 10.
   0.7 → 7: 1 place to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −1
5. Write the answer. Put A and the power of 10 together.
   7 × 10⁻¹

Check. Convert back: 7 × 10⁻¹ = 0.7 = 0.7. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.06 is 6.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 6
3. Count how many places the decimal point has moved. Starting from 0.06 the decimal point has to move 2 places to the right to sit after the 6. Each place moved is a factor of 10.
   0.06 → 6: 2 places to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −2
5. Write the answer. Put A and the power of 10 together.
   6 × 10⁻²

Check. Convert back: 6 × 10⁻² = 0.06 = 0.06. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.0035 is 3.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 3.5
3. Count how many places the decimal point has moved. Starting from 0.0035 the decimal point has to move 3 places to the right to sit after the 3. Each place moved is a factor of 10.
   0.0035 → 3.5: 3 places to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −3
5. Write the answer. Put A and the power of 10 together.
   3.5 × 10⁻³

Check. Convert back: 3.5 × 10⁻³ = 0.0035 = 0.0035. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.00072 is 7.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 7.2
3. Count how many places the decimal point has moved. Starting from 0.00072 the decimal point has to move 4 places to the right to sit after the 7. Each place moved is a factor of 10.
   0.00072 → 7.2: 4 places to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −4
5. Write the answer. Put A and the power of 10 together.
   7.2 × 10⁻⁴

Check. Convert back: 7.2 × 10⁻⁴ = 0.00072 = 0.00072. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.000081 is 8.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 8.1
3. Count how many places the decimal point has moved. Starting from 0.000081 the decimal point has to move 5 places to the right to sit after the 8. Each place moved is a factor of 10.
   0.000081 → 8.1: 5 places to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −5
5. Write the answer. Put A and the power of 10 together.
   8.1 × 10⁻⁵

Check. Convert back: 8.1 × 10⁻⁵ = 0.000081 = 0.000081. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.0000046 is 4.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 4.6
3. Count how many places the decimal point has moved. Starting from 0.0000046 the decimal point has to move 6 places to the right to sit after the 4. Each place moved is a factor of 10.
   0.0000046 → 4.6: 6 places to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −6
5. Write the answer. Put A and the power of 10 together.
   4.6 × 10⁻⁶

Check. Convert back: 4.6 × 10⁻⁶ = 0.0000046 = 0.0000046. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is 3, which is positive, so the number is big. Multiplying by 10 moves every digit one place to the left, which is the same as moving the decimal point one place to the right.
   10³ means multiply by 10, 3 times
2. Move the decimal point 3 places to the right. Move it one place at a time. If there are not enough digits, fill the gaps with zeros.
   5 → 50 → 500 → 5000
3. Write the answer. The last number in the chain is the answer.
   5000

Check. Convert back to standard form: 5000 has first non-zero digit 5, so A = 5 and n = 3. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is 4, which is positive, so the number is big. Multiplying by 10 moves every digit one place to the left, which is the same as moving the decimal point one place to the right.
   10⁴ means multiply by 10, 4 times
2. Move the decimal point 4 places to the right. Move it one place at a time. If there are not enough digits, fill the gaps with zeros.
   2.4 → 24 → 240 → 2400 → 24000
3. Write the answer. The last number in the chain is the answer.
   24,000

Check. Convert back to standard form: 24000 has first non-zero digit 2, so A = 2.4 and n = 4. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is 5, which is positive, so the number is big. Multiplying by 10 moves every digit one place to the left, which is the same as moving the decimal point one place to the right.
   10⁵ means multiply by 10, 5 times
2. Move the decimal point 5 places to the right. Move it one place at a time. If there are not enough digits, fill the gaps with zeros.
   7.08 → 70.8 → 708 → 7080 → 70800 → 708000
3. Write the answer. The last number in the chain is the answer.
   708,000

Check. Convert back to standard form: 708000 has first non-zero digit 7, so A = 7.08 and n = 5. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is 6, which is positive, so the number is big. Multiplying by 10 moves every digit one place to the left, which is the same as moving the decimal point one place to the right.
   10⁶ means multiply by 10, 6 times
2. Move the decimal point 6 places to the right. Move it one place at a time. If there are not enough digits, fill the gaps with zeros.
   1.5 → 15 → 150 → 1500 → 15000 → 150000 → 1500000
3. Write the answer. The last number in the chain is the answer.
   1,500,000

Check. Convert back to standard form: 1500000 has first non-zero digit 1, so A = 1.5 and n = 6. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is 7, which is positive, so the number is big. Multiplying by 10 moves every digit one place to the left, which is the same as moving the decimal point one place to the right.
   10⁷ means multiply by 10, 7 times
2. Move the decimal point 7 places to the right. Move it one place at a time. If there are not enough digits, fill the gaps with zeros.
   9.2 → 92 → 920 → 9200 → 92000 → 920000 → 9200000 → 92000000
3. Write the answer. The last number in the chain is the answer.
   92,000,000

Check. Convert back to standard form: 92000000 has first non-zero digit 9, so A = 9.2 and n = 7. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is −1, which is negative, so the number is small. Dividing by 10 moves every digit one place to the right, which is the same as moving the decimal point one place to the left.
   10⁻¹ means divide by 10, 1 times
2. Move the decimal point 1 place to the left. Move it one place at a time and fill the new places with zeros.
   4 → 0.4
3. Write the answer. The last number in the chain is the answer.
   0.4

Check. Convert back to standard form: 0.4 has first non-zero digit 4, so A = 4 and n = −1. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is −3, which is negative, so the number is small. Dividing by 10 moves every digit one place to the right, which is the same as moving the decimal point one place to the left.
   10⁻³ means divide by 10, 3 times
2. Move the decimal point 3 places to the left. Move it one place at a time and fill the new places with zeros.
   6.2 → 0.62 → 0.062 → 0.0062
3. Write the answer. The last number in the chain is the answer.
   0.0062

Check. Convert back to standard form: 0.0062 has first non-zero digit 6, so A = 6.2 and n = −3. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is −4, which is negative, so the number is small. Dividing by 10 moves every digit one place to the right, which is the same as moving the decimal point one place to the left.
   10⁻⁴ means divide by 10, 4 times
2. Move the decimal point 4 places to the left. Move it one place at a time and fill the new places with zeros.
   3.07 → 0.307 → 0.0307 → 0.00307 → 0.000307
3. Write the answer. The last number in the chain is the answer.
   0.000307

Check. Convert back to standard form: 0.000307 has first non-zero digit 3, so A = 3.07 and n = −4. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is −5, which is negative, so the number is small. Dividing by 10 moves every digit one place to the right, which is the same as moving the decimal point one place to the left.
   10⁻⁵ means divide by 10, 5 times
2. Move the decimal point 5 places to the left. Move it one place at a time and fill the new places with zeros.
   8.5 → 0.85 → 0.085 → 0.0085 → 0.00085 → 0.000085
3. Write the answer. The last number in the chain is the answer.
   0.000085

Check. Convert back to standard form: 0.000085 has first non-zero digit 8, so A = 8.5 and n = −5. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is −6, which is negative, so the number is small. Dividing by 10 moves every digit one place to the right, which is the same as moving the decimal point one place to the left.
   10⁻⁶ means divide by 10, 6 times
2. Move the decimal point 6 places to the left. Move it one place at a time and fill the new places with zeros.
   1.1 → 0.11 → 0.011 → 0.0011 → 0.00011 → 0.000011 → 0.0000011
3. Write the answer. The last number in the chain is the answer.
   0.0000011

Check. Convert back to standard form: 0.0000011 has first non-zero digit 1, so A = 1.1 and n = −6. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Change the front number A so that it is between 1 and 10, and adjust the power of 10 to match.

1. Check the rule for standard form. In A × 10ⁿ, A must be at least 1 and less than 10.
   A = 45, which is 10 or more, so this is not in standard form.
2. Write 45 as a number between 1 and 10 times a power of 10. Moving the decimal point 1 place to the left makes the number 10 times smaller, so we must multiply by 10¹ to keep the value the same.
   45 = 4.5 × 10¹
3. Replace A in the original and combine the powers of 10. Powers of 10 multiply by adding the indices: 10^a × 10^b = 10^a+b.
   45 × 10³ = 4.5 × 10¹ × 10³
   = 4.5 × 10^{1 + 3}
4. Write the answer.
   4.5 × 10⁴

Check. Both forms are the same number, 45000: 45 × 1000 = 45000 and 4.5 × 10000 = 45000. ✓

Watch out. If you make A smaller by moving the point left, the power of 10 gets bigger by the same number of places (and the other way round).

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Change the front number A so that it is between 1 and 10, and adjust the power of 10 to match.

1. Check the rule for standard form. In A × 10ⁿ, A must be at least 1 and less than 10.
   A = 0.6, which is less than 1, so this is not in standard form.
2. Write 0.6 as a number between 1 and 10 times a power of 10. Moving the decimal point 1 place to the right makes the number 10 times bigger, so we must multiply by 10⁻¹ (that is, divide by 10) to keep the value the same.
   0.6 = 6 × 10⁻¹
3. Replace A in the original and combine the powers of 10. Powers of 10 multiply by adding the indices: 10^a × 10^b = 10^a+b.
   0.6 × 10⁵ = 6 × 10⁻¹ × 10⁵
   = 6 × 10^{−1 + 5}
4. Write the answer.
   6 × 10⁴

Check. Both forms are the same number, 60000: 0.6 × 100000 = 60000 and 6 × 10000 = 60000. ✓

Watch out. If you make A smaller by moving the point left, the power of 10 gets bigger by the same number of places (and the other way round).

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Change the front number A so that it is between 1 and 10, and adjust the power of 10 to match.

1. Check the rule for standard form. In A × 10ⁿ, A must be at least 1 and less than 10.
   A = 120, which is 10 or more, so this is not in standard form.
2. Write 120 as a number between 1 and 10 times a power of 10. Moving the decimal point 2 places to the left makes the number 100 times smaller, so we must multiply by 10² to keep the value the same.
   120 = 1.2 × 10²
3. Replace A in the original and combine the powers of 10. Powers of 10 multiply by adding the indices: 10^a × 10^b = 10^a+b.
   120 × 10⁻⁴ = 1.2 × 10² × 10⁻⁴
   = 1.2 × 10^{2 + (−4)}
4. Write the answer.
   1.2 × 10⁻²

Check. Both forms are the same number, 0.012: 120 × 0.0001 = 0.012 and 1.2 × 0.01 = 0.012. ✓

Watch out. If you make A smaller by moving the point left, the power of 10 gets bigger by the same number of places (and the other way round).

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Change the front number A so that it is between 1 and 10, and adjust the power of 10 to match.

1. Check the rule for standard form. In A × 10ⁿ, A must be at least 1 and less than 10.
   A = 0.08, which is less than 1, so this is not in standard form.
2. Write 0.08 as a number between 1 and 10 times a power of 10. Moving the decimal point 2 places to the right makes the number 100 times bigger, so we must multiply by 10⁻² (that is, divide by 100) to keep the value the same.
   0.08 = 8 × 10⁻²
3. Replace A in the original and combine the powers of 10. Powers of 10 multiply by adding the indices: 10^a × 10^b = 10^a+b.
   0.08 × 10⁷ = 8 × 10⁻² × 10⁷
   = 8 × 10^{−2 + 7}
4. Write the answer.
   8 × 10⁵

Check. Both forms are the same number, 800000: 0.08 × 10000000 = 800000 and 8 × 100000 = 800000. ✓

Watch out. If you make A smaller by moving the point left, the power of 10 gets bigger by the same number of places (and the other way round).

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Change the front number A so that it is between 1 and 10, and adjust the power of 10 to match.

1. Check the rule for standard form. In A × 10ⁿ, A must be at least 1 and less than 10.
   A = 250, which is 10 or more, so this is not in standard form.
2. Write 250 as a number between 1 and 10 times a power of 10. Moving the decimal point 2 places to the left makes the number 100 times smaller, so we must multiply by 10² to keep the value the same.
   250 = 2.5 × 10²
3. Replace A in the original and combine the powers of 10. Powers of 10 multiply by adding the indices: 10^a × 10^b = 10^a+b.
   250 × 10² = 2.5 × 10² × 10²
   = 2.5 × 10^{2 + 2}
4. Write the answer.
   2.5 × 10⁴

Check. Both forms are the same number, 25000: 250 × 100 = 25000 and 2.5 × 10000 = 25000. ✓

Watch out. If you make A smaller by moving the point left, the power of 10 gets bigger by the same number of places (and the other way round).

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. The E on a calculator display stands for "times ten to the power of". The display is standard form.

1. Read the calculator display. The letter E means "× 10 to the power". The number before E is A. The number after E is the power of 10.
   3.6E8 means 3.6 × 10⁸
2. Write the ordinary number. Move the decimal point 8 places to the right.
   360,000,000

Check. Back to standard form: the first non-zero digit of 360000000 is 3, giving 3.6 × 10⁸. ✓

Watch out. E5 does not mean 5. 3.6E8 is 3.6 × 10⁸.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. The E on a calculator display stands for "times ten to the power of". The display is standard form.

1. Read the calculator display. The letter E means "× 10 to the power". The number before E is A. The number after E is the power of 10.
   2.4E-3 means 2.4 × 10⁻³
2. Write the ordinary number. Move the decimal point 3 places to the left.
   0.0024

Check. Back to standard form: the first non-zero digit of 0.0024 is 2, giving 2.4 × 10⁻³. ✓

Watch out. E5 does not mean 5. 3.6E8 is 3.6 × 10⁸.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. The E on a calculator display stands for "times ten to the power of". The display is standard form.

1. Read the calculator display. The letter E means "× 10 to the power". The number before E is A. The number after E is the power of 10.
   7.1E5 means 7.1 × 10⁵
2. Write the ordinary number. Move the decimal point 5 places to the right.
   710,000

Check. Back to standard form: the first non-zero digit of 710000 is 7, giving 7.1 × 10⁵. ✓

Watch out. E5 does not mean 5. 3.6E8 is 3.6 × 10⁸.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. The E on a calculator display stands for "times ten to the power of". The display is standard form.

1. Read the calculator display. The letter E means "× 10 to the power". The number before E is A. The number after E is the power of 10.
   5E-2 means 5 × 10⁻²
2. Write the ordinary number. Move the decimal point 2 places to the left.
   0.05

Check. Back to standard form: the first non-zero digit of 0.05 is 5, giving 5 × 10⁻². ✓

Watch out. E5 does not mean 5. 3.6E8 is 3.6 × 10⁸.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Round to 2 significant figures first. Significant figures count from the first non-zero digit.
   4,523,000 = 4,500,000 to 2 s.f.
2. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 4,500,000 is 4.
3. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 4.5
4. Count how many places the decimal point has moved. Starting from 4500000 the decimal point has to move 6 places to the left to sit after the 4. Each place moved is a factor of 10.
   4500000 → 4.5: 6 places to the left
5. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 6
6. Write the answer. Put A and the power of 10 together.
   4.5 × 10⁶

Check. Convert back: 4.5 × 10⁶ = 4500000 (which is 4,500,000 to 2 s.f.). ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Round to 2 significant figures first. Significant figures count from the first non-zero digit.
   0.0003846 = 0.00038 to 2 s.f.
2. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.00038 is 3.
3. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 3.8
4. Count how many places the decimal point has moved. Starting from 0.00038 the decimal point has to move 4 places to the right to sit after the 3. Each place moved is a factor of 10.
   0.00038 → 3.8: 4 places to the right
5. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −4
6. Write the answer. Put A and the power of 10 together.
   3.8 × 10⁻⁴

Check. Convert back: 3.8 × 10⁻⁴ = 0.00038 (which is 0.00038 to 2 s.f.). ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Round to 3 significant figures first. Significant figures count from the first non-zero digit.
   299,792,458 = 300,000,000 to 3 s.f.
2. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 300,000,000 is 3.
3. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 3.00
4. Count how many places the decimal point has moved. Starting from 300000000 the decimal point has to move 8 places to the left to sit after the 3. Each place moved is a factor of 10.
   300000000 → 3.00: 8 places to the left
5. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 8
6. Write the answer. Put A and the power of 10 together.
   3.00 × 10⁸

Check. Convert back: 3.00 × 10⁸ = 300000000 (which is 300,000,000 to 3 s.f.). ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Round to 3 significant figures first. Significant figures count from the first non-zero digit.
   6,371,000 = 6,370,000 to 3 s.f.
2. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 6,370,000 is 6.
3. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 6.37
4. Count how many places the decimal point has moved. Starting from 6370000 the decimal point has to move 6 places to the left to sit after the 6. Each place moved is a factor of 10.
   6370000 → 6.37: 6 places to the left
5. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 6
6. Write the answer. Put A and the power of 10 together.
   6.37 × 10⁶

Check. Convert back: 6.37 × 10⁶ = 6370000 (which is 6,370,000 to 3 s.f.). ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 8,100,000,000 is 8.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 8.1
3. Count how many places the decimal point has moved. Starting from 8100000000 the decimal point has to move 9 places to the left to sit after the 8. Each place moved is a factor of 10.
   8100000000 → 8.1: 9 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 9
5. Write the answer. Put A and the power of 10 together.
   8.1 × 10⁹

Check. Convert back: 8.1 × 10⁹ = 8100000000 = 8,100,000,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 384,400 is 3.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 3.844
3. Count how many places the decimal point has moved. Starting from 384400 the decimal point has to move 5 places to the left to sit after the 3. Each place moved is a factor of 10.
   384400 → 3.844: 5 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 5
5. Write the answer. Put A and the power of 10 together.
   3.844 × 10⁵

Check. Convert back: 3.844 × 10⁵ = 384400 = 384,400. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.0001 is 1.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 1
3. Count how many places the decimal point has moved. Starting from 0.0001 the decimal point has to move 4 places to the right to sit after the 1. Each place moved is a factor of 10.
   0.0001 → 1: 4 places to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −4
5. Write the answer. Put A and the power of 10 together.
   1 × 10⁻⁴

Check. Convert back: 1 × 10⁻⁴ = 0.0001 = 0.0001. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.00000000000000000000000000167 is 1.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 1.67
3. Count how many places the decimal point has moved. Starting from 0.00000000000000000000000000167 the decimal point has to move 27 places to the right to sit after the 1. Each place moved is a factor of 10.
   0.00000000000000000000000000167 → 1.67: 27 places to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −27
5. Write the answer. Put A and the power of 10 together.
   1.67 × 10⁻²⁷

Check. Convert back: 1.67 × 10⁻²⁷ = 0.00000000000000000000000000167 = 0.00000000000000000000000000167. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 25,000,000 is 2.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 2.5
3. Count how many places the decimal point has moved. Starting from 25000000 the decimal point has to move 7 places to the left to sit after the 2. Each place moved is a factor of 10.
   25000000 → 2.5: 7 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 7
5. Write the answer. Put A and the power of 10 together.
   2.5 × 10⁷

Check. Convert back: 2.5 × 10⁷ = 25000000 = 25,000,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 500,000,000 is 5.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 5
3. Count how many places the decimal point has moved. Starting from 500000000 the decimal point has to move 8 places to the left to sit after the 5. Each place moved is a factor of 10.
   500000000 → 5: 8 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 8
5. Write the answer. Put A and the power of 10 together.
   5 × 10⁸

Check. Convert back: 5 × 10⁸ = 500000000 = 500,000,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.003 is 3.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 3
3. Count how many places the decimal point has moved. Starting from 0.003 the decimal point has to move 3 places to the right to sit after the 3. Each place moved is a factor of 10.
   0.003 → 3: 3 places to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −3
5. Write the answer. Put A and the power of 10 together.
   3 × 10⁻³

Check. Convert back: 3 × 10⁻³ = 0.003 = 0.003. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.00001 is 1.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 1
3. Count how many places the decimal point has moved. Starting from 0.00001 the decimal point has to move 5 places to the right to sit after the 1. Each place moved is a factor of 10.
   0.00001 → 1: 5 places to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −5
5. Write the answer. Put A and the power of 10 together.
   1 × 10⁻⁵

Check. Convert back: 1 × 10⁻⁵ = 0.00001 = 0.00001. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Multiplying by 10 adds 1 to the power.

1. Write 10N.
   10 × 4.5 × 10⁶
2. Combine the powers of 10. 10¹ × 10⁶ = 10⁷.
   = 4.5 × 10¹⁺⁶
3. Write the answer.
   4.5 × 10⁷

Check. 4,500,000 × 10 = 45,000,000 = 4.5 × 10⁷. ✓

Watch out. The front number does not change.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Dividing by 100 subtracts 2 from the power.

1. Write the division. 100 = 10².
   4.5 × 10⁶ ÷ 10²
2. Subtract the indices. 10⁶ ÷ 10² = 10⁶⁻².
   = 4.5 × 10⁴
3. Write the answer.
   4.5 × 10⁴

Check. 4,500,000 ÷ 100 = 45,000 = 4.5 × 10⁴. ✓

Watch out. Dividing by 100 lowers the power by 2.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 1,392,000 is 1.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 1.392
3. Count how many places the decimal point has moved. Starting from 1392000 the decimal point has to move 6 places to the left to sit after the 1. Each place moved is a factor of 10.
   1392000 → 1.392: 6 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 6
5. Write the answer. Put A and the power of 10 together.
   1.392 × 10⁶

Check. Convert back: 1.392 × 10⁶ = 1392000 = 1,392,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.0000007 is 7.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 7
3. Count how many places the decimal point has moved. Starting from 0.0000007 the decimal point has to move 7 places to the right to sit after the 7. Each place moved is a factor of 10.
   0.0000007 → 7: 7 places to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −7
5. Write the answer. Put A and the power of 10 together.
   7 × 10⁻⁷

Check. Convert back: 7 × 10⁻⁷ = 0.0000007 = 0.0000007. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 67,000,000 is 6.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 6.7
3. Count how many places the decimal point has moved. Starting from 67000000 the decimal point has to move 7 places to the left to sit after the 6. Each place moved is a factor of 10.
   67000000 → 6.7: 7 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 7
5. Write the answer. Put A and the power of 10 together.
   6.7 × 10⁷

Check. Convert back: 6.7 × 10⁷ = 67000000 = 67,000,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.00007 is 7.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 7
3. Count how many places the decimal point has moved. Starting from 0.00007 the decimal point has to move 5 places to the right to sit after the 7. Each place moved is a factor of 10.
   0.00007 → 7: 5 places to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −5
5. Write the answer. Put A and the power of 10 together.
   7 × 10⁻⁵

Check. Convert back: 7 × 10⁻⁵ = 0.00007 = 0.00007. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 40,000,000,000,000 is 4.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 4
3. Count how many places the decimal point has moved. Starting from 40000000000000 the decimal point has to move 13 places to the left to sit after the 4. Each place moved is a factor of 10.
   40000000000000 → 4: 13 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 13
5. Write the answer. Put A and the power of 10 together.
   4 × 10¹³

Check. Convert back: 4 × 10¹³ = 40000000000000 = 40,000,000,000,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is 8, which is positive, so the number is big. Multiplying by 10 moves every digit one place to the left, which is the same as moving the decimal point one place to the right.
   10⁸ means multiply by 10, 8 times
2. Move the decimal point 8 places to the right. Move it one place at a time. If there are not enough digits, fill the gaps with zeros.
   1.5 → 15 → 150 → 1500 → 15000 → 150000 → 1500000 → 15000000 → 150000000
3. Write the answer. The last number in the chain is the answer.
   150,000,000

Check. Convert back to standard form: 150000000 has first non-zero digit 1, so A = 1.5 and n = 8. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is −7, which is negative, so the number is small. Dividing by 10 moves every digit one place to the right, which is the same as moving the decimal point one place to the left.
   10⁻⁷ means divide by 10, 7 times
2. Move the decimal point 7 places to the left. Move it one place at a time and fill the new places with zeros.
   1.2 → 0.12 → 0.012 → 0.0012 → 0.00012 → 0.000012 → 0.0000012 → 0.00000012
3. Write the answer. The last number in the chain is the answer.
   0.00000012

Check. Convert back to standard form: 0.00000012 has first non-zero digit 1, so A = 1.2 and n = −7. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is 9, which is positive, so the number is big. Multiplying by 10 moves every digit one place to the left, which is the same as moving the decimal point one place to the right.
   10⁹ means multiply by 10, 9 times
2. Move the decimal point 9 places to the right. Move it one place at a time. If there are not enough digits, fill the gaps with zeros.
   3 → 30 → 300 → 3000 → 30000 → 300000 → 3000000 → 30000000 → 300000000 → 3000000000
3. Write the answer. The last number in the chain is the answer.
   3,000,000,000

Check. Convert back to standard form: 3000000000 has first non-zero digit 3, so A = 3 and n = 9. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is −9, which is negative, so the number is small. Dividing by 10 moves every digit one place to the right, which is the same as moving the decimal point one place to the left.
   10⁻⁹ means divide by 10, 9 times
2. Move the decimal point 9 places to the left. Move it one place at a time and fill the new places with zeros.
   2.5 → 0.25 → 0.025 → 0.0025 → 0.00025 → 0.000025 → 0.0000025 → 0.00000025 → 0.000000025 → 0.0000000025
3. Write the answer. The last number in the chain is the answer.
   0.0000000025

Check. Convert back to standard form: 0.0000000025 has first non-zero digit 2, so A = 2.5 and n = −9. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 85,000 is 8.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 8.5
3. Count how many places the decimal point has moved. Starting from 85000 the decimal point has to move 4 places to the left to sit after the 8. Each place moved is a factor of 10.
   85000 → 8.5: 4 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 4
5. Write the answer. Put A and the power of 10 together.
   8.5 × 10⁴

Check. Convert back: 8.5 × 10⁴ = 85000 = 85,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.0092 is 9.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 9.2
3. Count how many places the decimal point has moved. Starting from 0.0092 the decimal point has to move 3 places to the right to sit after the 9. Each place moved is a factor of 10.
   0.0092 → 9.2: 3 places to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −3
5. Write the answer. Put A and the power of 10 together.
   9.2 × 10⁻³

Check. Convert back: 9.2 × 10⁻³ = 0.0092 = 0.0092. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 630,000,000 is 6.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 6.3
3. Count how many places the decimal point has moved. Starting from 630000000 the decimal point has to move 8 places to the left to sit after the 6. Each place moved is a factor of 10.
   630000000 → 6.3: 8 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 8
5. Write the answer. Put A and the power of 10 together.
   6.3 × 10⁸

Check. Convert back: 6.3 × 10⁸ = 630000000 = 630,000,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 0.00000058 is 5.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 5.8
3. Count how many places the decimal point has moved. Starting from 0.00000058 the decimal point has to move 7 places to the right to sit after the 5. Each place moved is a factor of 10.
   0.00000058 → 5.8: 7 places to the right
4. Decide the sign of the power. The number is less than 1 (a small number). We made it bigger to get A, so we must divide by 10 to put the size back. Dividing by 10 is the same as multiplying by 10⁻¹.
   n is negative: n = −7
5. Write the answer. Put A and the power of 10 together.
   5.8 × 10⁻⁷

Check. Convert back: 5.8 × 10⁻⁷ = 0.00000058 = 0.00000058. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Write the number as A × 10ⁿ: put the decimal point after the first non-zero digit, then count how many places it moved.

1. Find the first non-zero digit. In standard form A is at least 1 and less than 10, so A has exactly one non-zero digit in front of the decimal point. That digit is the first non-zero digit of the number.
   The first non-zero digit of 1,200,000,000 is 1.
2. Place the decimal point just after it to make A. The digits of A are the same as the digits of the number (without the zeros at the ends).
   A = 1.2
3. Count how many places the decimal point has moved. Starting from 1200000000 the decimal point has to move 9 places to the left to sit after the 1. Each place moved is a factor of 10.
   1200000000 → 1.2: 9 places to the left
4. Decide the sign of the power. The number is 10 or more (a big number). We made it smaller to get A, so we must multiply by 10 to put the size back.
   n is positive: n = 9
5. Write the answer. Put A and the power of 10 together.
   1.2 × 10⁹

Check. Convert back: 1.2 × 10⁹ = 1200000000 = 1,200,000,000. ✓

Watch out. A big number has a positive power and a small number (less than 1) has a negative power. Do not count the zeros, count the places the point moves.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is 4, which is positive, so the number is big. Multiplying by 10 moves every digit one place to the left, which is the same as moving the decimal point one place to the right.
   10⁴ means multiply by 10, 4 times
2. Move the decimal point 4 places to the right. Move it one place at a time. If there are not enough digits, fill the gaps with zeros.
   6.3 → 63 → 630 → 6300 → 63000
3. Write the answer. The last number in the chain is the answer.
   63,000

Check. Convert back to standard form: 63000 has first non-zero digit 6, so A = 6.3 and n = 4. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is −3, which is negative, so the number is small. Dividing by 10 moves every digit one place to the right, which is the same as moving the decimal point one place to the left.
   10⁻³ means divide by 10, 3 times
2. Move the decimal point 3 places to the left. Move it one place at a time and fill the new places with zeros.
   9 → 0.9 → 0.09 → 0.009
3. Write the answer. The last number in the chain is the answer.
   0.009

Check. Convert back to standard form: 0.009 has first non-zero digit 9, so A = 9 and n = −3. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is 6, which is positive, so the number is big. Multiplying by 10 moves every digit one place to the left, which is the same as moving the decimal point one place to the right.
   10⁶ means multiply by 10, 6 times
2. Move the decimal point 6 places to the right. Move it one place at a time. If there are not enough digits, fill the gaps with zeros.
   1.07 → 10.7 → 107 → 1070 → 10700 → 107000 → 1070000
3. Write the answer. The last number in the chain is the answer.
   1,070,000

Check. Convert back to standard form: 1070000 has first non-zero digit 1, so A = 1.07 and n = 6. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. A positive power of 10 makes the number bigger (move the decimal point right). A negative power makes it smaller (move the decimal point left).

1. Look at the power of 10. The power is −2, which is negative, so the number is small. Dividing by 10 moves every digit one place to the right, which is the same as moving the decimal point one place to the left.
   10⁻² means divide by 10, 2 times
2. Move the decimal point 2 places to the left. Move it one place at a time and fill the new places with zeros.
   4.4 → 0.44 → 0.044
3. Write the answer. The last number in the chain is the answer.
   0.044

Check. Convert back to standard form: 0.044 has first non-zero digit 4, so A = 4.4 and n = −2. ✓

Watch out. A negative power makes a number smaller, but it does not make it negative.

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Change the front number A so that it is between 1 and 10, and adjust the power of 10 to match.

1. Check the rule for standard form. In A × 10ⁿ, A must be at least 1 and less than 10.
   A = 32, which is 10 or more, so this is not in standard form.
2. Write 32 as a number between 1 and 10 times a power of 10. Moving the decimal point 1 place to the left makes the number 10 times smaller, so we must multiply by 10¹ to keep the value the same.
   32 = 3.2 × 10¹
3. Replace A in the original and combine the powers of 10. Powers of 10 multiply by adding the indices: 10^a × 10^b = 10^a+b.
   32 × 10⁶ = 3.2 × 10¹ × 10⁶
   = 3.2 × 10^{1 + 6}
4. Write the answer.
   3.2 × 10⁷

Check. Both forms are the same number, 32000000: 32 × 1000000 = 32000000 and 3.2 × 10000000 = 32000000. ✓

Watch out. If you make A smaller by moving the point left, the power of 10 gets bigger by the same number of places (and the other way round).

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Change the front number A so that it is between 1 and 10, and adjust the power of 10 to match.

1. Check the rule for standard form. In A × 10ⁿ, A must be at least 1 and less than 10.
   A = 0.45, which is less than 1, so this is not in standard form.
2. Write 0.45 as a number between 1 and 10 times a power of 10. Moving the decimal point 1 place to the right makes the number 10 times bigger, so we must multiply by 10⁻¹ (that is, divide by 10) to keep the value the same.
   0.45 = 4.5 × 10⁻¹
3. Replace A in the original and combine the powers of 10. Powers of 10 multiply by adding the indices: 10^a × 10^b = 10^a+b.
   0.45 × 10³ = 4.5 × 10⁻¹ × 10³
   = 4.5 × 10^{−1 + 3}
4. Write the answer.
   4.5 × 10²

Check. Both forms are the same number, 450: 0.45 × 1000 = 450 and 4.5 × 100 = 450. ✓

Watch out. If you make A smaller by moving the point left, the power of 10 gets bigger by the same number of places (and the other way round).

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Change the front number A so that it is between 1 and 10, and adjust the power of 10 to match.

1. Check the rule for standard form. In A × 10ⁿ, A must be at least 1 and less than 10.
   A = 560, which is 10 or more, so this is not in standard form.
2. Write 560 as a number between 1 and 10 times a power of 10. Moving the decimal point 2 places to the left makes the number 100 times smaller, so we must multiply by 10² to keep the value the same.
   560 = 5.6 × 10²
3. Replace A in the original and combine the powers of 10. Powers of 10 multiply by adding the indices: 10^a × 10^b = 10^a+b.
   560 × 10⁻⁵ = 5.6 × 10² × 10⁻⁵
   = 5.6 × 10^{2 + (−5)}
4. Write the answer.
   5.6 × 10⁻³

Check. Both forms are the same number, 0.0056: 560 × 0.00001 = 0.0056 and 5.6 × 0.001 = 0.0056. ✓

Watch out. If you make A smaller by moving the point left, the power of 10 gets bigger by the same number of places (and the other way round).

Key words. Standard form is a way to write very big or very small numbers as A × 10ⁿ, where A is at least 1 and less than 10 (so exactly one non-zero digit before the decimal point) and n is a whole number (positive, negative or zero).

Method. Change the front number A so that it is between 1 and 10, and adjust the power of 10 to match.

1. Check the rule for standard form. In A × 10ⁿ, A must be at least 1 and less than 10.
   A = 0.09, which is less than 1, so this is not in standard form.
2. Write 0.09 as a number between 1 and 10 times a power of 10. Moving the decimal point 2 places to the right makes the number 100 times bigger, so we must multiply by 10⁻² (that is, divide by 100) to keep the value the same.
   0.09 = 9 × 10⁻²
3. Replace A in the original and combine the powers of 10. Powers of 10 multiply by adding the indices: 10^a × 10^b = 10^a+b.
   0.09 × 10⁴ = 9 × 10⁻² × 10⁴
   = 9 × 10^{−2 + 4}
4. Write the answer.
   9 × 10²

Check. Both forms are the same number, 900: 0.09 × 10000 = 900 and 9 × 100 = 900. ✓

Watch out. If you make A smaller by moving the point left, the power of 10 gets bigger by the same number of places (and the other way round).