GCSE Maths: Comparing and calculating with standard form (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. Compare the powers of 10 first. If the powers are the same, compare the front numbers.

1. Check both numbers are in standard form. The front numbers must be between 1 and 10 for this comparison to work.
   3.2 and 9.9 are both at least 1 and less than 10.
2. Compare the powers of 10 first. A bigger power of 10 means a bigger number, because both front numbers are between 1 and 10 and so cannot make up the difference.
   10⁵ and 10⁴: 5 > 4
3. Decide. The powers are different, so the powers decide it. We do not need to look at the front numbers.
   3.2 × 10⁵ > 9.9 × 10⁴

Check. As ordinary numbers: 320000 and 99000, and 320000 is bigger than 99000. ✓

Watch out. Do not compare the front numbers first. 9.9 × 10² is much smaller than 1.1 × 10⁵.

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. Compare the powers of 10 first. If the powers are the same, compare the front numbers.

1. Check both numbers are in standard form. The front numbers must be between 1 and 10 for this comparison to work.
   4.5 and 4.5 are both at least 1 and less than 10.
2. Compare the powers of 10 first. A bigger power of 10 means a bigger number, because both front numbers are between 1 and 10 and so cannot make up the difference. For negative powers remember that −3 and −5 are numbers on a number line: the one further to the right is bigger. For example 10⁻² = 0.01 is bigger than 10⁻⁵ = 0.00001.
   10⁻³ and 10⁻⁵: −3 > −5
3. Decide. The powers are different, so the powers decide it. We do not need to look at the front numbers.
   4.5 × 10⁻³ > 4.5 × 10⁻⁵

Check. As ordinary numbers: 0.0045 and 0.000045, and 0.0045 is bigger than 0.000045. ✓

Watch out. Do not compare the front numbers first. 9.9 × 10² is much smaller than 1.1 × 10⁵.

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. Compare the powers of 10 first. If the powers are the same, compare the front numbers.

1. Check both numbers are in standard form. The front numbers must be between 1 and 10 for this comparison to work.
   6.1 and 6.8 are both at least 1 and less than 10.
2. Compare the powers of 10 first. If the powers are equal, the front numbers decide.
   Both powers are 10⁷: they are the same.
3. Compare the front numbers. With the same power of 10, the bigger A gives the bigger number.
   6.1 < 6.8, so 6.1 × 10⁷ < 6.8 × 10⁷

Check. As ordinary numbers: 61000000 and 68000000. ✓

Watch out. Do not compare the front numbers first. 9.9 × 10² is much smaller than 1.1 × 10⁵.

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. Compare the powers of 10 first. If the powers are the same, compare the front numbers.

1. Check both numbers are in standard form. The front numbers must be between 1 and 10 for this comparison to work.
   2 and 9 are both at least 1 and less than 10.
2. Compare the powers of 10 first. A bigger power of 10 means a bigger number, because both front numbers are between 1 and 10 and so cannot make up the difference. For negative powers remember that −6 and −7 are numbers on a number line: the one further to the right is bigger. For example 10⁻² = 0.01 is bigger than 10⁻⁵ = 0.00001.
   10⁻⁶ and 10⁻⁷: −6 > −7
3. Decide. The powers are different, so the powers decide it. We do not need to look at the front numbers.
   2 × 10⁻⁶ > 9 × 10⁻⁷

Check. As ordinary numbers: 0.000002 and 0.0000009, and 0.000002 is bigger than 0.0000009. ✓

Watch out. Do not compare the front numbers first. 9.9 × 10² is much smaller than 1.1 × 10⁵.

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. Compare the powers of 10 first. If the powers are the same, compare the front numbers.

1. Check both numbers are in standard form. The front numbers must be between 1 and 10 for this comparison to work.
   1.2 and 9.9 are both at least 1 and less than 10.
2. Compare the powers of 10 first. A bigger power of 10 means a bigger number, because both front numbers are between 1 and 10 and so cannot make up the difference.
   10¹⁰ and 10⁹: 10 > 9
3. Decide. The powers are different, so the powers decide it. We do not need to look at the front numbers.
   1.2 × 10¹⁰ > 9.9 × 10⁹

Check. The power 10 is bigger than 9. ✓

Watch out. Do not compare the front numbers first. 9.9 × 10² is much smaller than 1.1 × 10⁵.

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. Compare the powers of 10 first. If the powers are the same, compare the front numbers.

1. Check both numbers are in standard form. The front numbers must be between 1 and 10 for this comparison to work.
   7.5 and 2.1 are both at least 1 and less than 10.
2. Compare the powers of 10 first. A bigger power of 10 means a bigger number, because both front numbers are between 1 and 10 and so cannot make up the difference.
   10³ and 10⁶: 3 < 6
3. Decide. The powers are different, so the powers decide it. We do not need to look at the front numbers.
   7.5 × 10³ < 2.1 × 10⁶

Check. As ordinary numbers: 7500 and 2100000, and 7500 is smaller than 2100000. ✓

Watch out. Do not compare the front numbers first. 9.9 × 10² is much smaller than 1.1 × 10⁵.

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. Compare the powers of 10 first. If the powers are the same, compare the front numbers.

1. Check both numbers are in standard form. The front numbers must be between 1 and 10 for this comparison to work.
   3.3 and 1.9 are both at least 1 and less than 10.
2. Compare the powers of 10 first. A bigger power of 10 means a bigger number, because both front numbers are between 1 and 10 and so cannot make up the difference. For negative powers remember that −2 and −4 are numbers on a number line: the one further to the right is bigger. For example 10⁻² = 0.01 is bigger than 10⁻⁵ = 0.00001.
   10⁻² and 10⁻⁴: −2 > −4
3. Decide. The powers are different, so the powers decide it. We do not need to look at the front numbers.
   3.3 × 10⁻² > 1.9 × 10⁻⁴

Check. As ordinary numbers: 0.033 and 0.00019, and 0.033 is bigger than 0.00019. ✓

Watch out. Do not compare the front numbers first. 9.9 × 10² is much smaller than 1.1 × 10⁵.

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. Compare the powers of 10 first. If the powers are the same, compare the front numbers.

1. Check both numbers are in standard form. The front numbers must be between 1 and 10 for this comparison to work.
   5.6 and 5.1 are both at least 1 and less than 10.
2. Compare the powers of 10 first. If the powers are equal, the front numbers decide.
   Both powers are 10⁸: they are the same.
3. Compare the front numbers. With the same power of 10, the bigger A gives the bigger number.
   5.6 > 5.1, so 5.6 × 10⁸ > 5.1 × 10⁸

Check. As ordinary numbers: 560000000 and 510000000. ✓

Watch out. Do not compare the front numbers first. 9.9 × 10² is much smaller than 1.1 × 10⁵.

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. Compare the powers of 10 first. If the powers are the same, compare the front numbers.

1. Check both numbers are in standard form. The front numbers must be between 1 and 10 for this comparison to work.
   9 and 1 are both at least 1 and less than 10.
2. Compare the powers of 10 first. A bigger power of 10 means a bigger number, because both front numbers are between 1 and 10 and so cannot make up the difference. For negative powers remember that −5 and −4 are numbers on a number line: the one further to the right is bigger. For example 10⁻² = 0.01 is bigger than 10⁻⁵ = 0.00001.
   10⁻⁵ and 10⁻⁴: −5 < −4
3. Decide. The powers are different, so the powers decide it. We do not need to look at the front numbers.
   9 × 10⁻⁵ < 1 × 10⁻⁴

Check. As ordinary numbers: 0.00009 and 0.0001, and 0.00009 is smaller than 0.0001. ✓

Watch out. Do not compare the front numbers first. 9.9 × 10² is much smaller than 1.1 × 10⁵.

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. Order by the power of 10 first; use the front number to break ties.

1. List the power of 10 of each number. The power of 10 is the main thing that decides the size.
   2.5 × 10⁴: power 4
   7.1 × 10⁻³: power −3
   2.1 × 10⁴: power 4
   9 × 10³: power 3
2. Order by the powers of 10. We need smallest to largest. Remember that −7 is smaller than −3.
   power −3, power 3, power 4, power 4
3. If two numbers have the same power, compare their front numbers. With equal powers, the bigger front number is the bigger number.
   power 4: 2.5 and 2.1 → smaller first is 2.1
4. Write the numbers in order.
   7.1 × 10⁻³ ,  9 × 10³ ,  2.1 × 10⁴ ,  2.5 × 10⁴

Check. As ordinary numbers: 0.0071, 9000, 21000, 25000 — these increase along the list. ✓

Watch out. A negative power is smaller than a positive power, and a larger negative power (like −8) is smaller than −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. Order by the power of 10 first; use the front number to break ties.

1. List the power of 10 of each number. The power of 10 is the main thing that decides the size.
   5.2 × 10⁻⁶: power −6
   4.9 × 10⁻⁵: power −5
   1.1 × 10⁻⁵: power −5
   8 × 10⁻⁷: power −7
2. Order by the powers of 10. We need smallest to largest. Remember that −7 is smaller than −3.
   power −7, power −6, power −5, power −5
3. If two numbers have the same power, compare their front numbers. With equal powers, the bigger front number is the bigger number.
   power −5: 4.9 and 1.1 → smaller first is 1.1
4. Write the numbers in order.
   8 × 10⁻⁷ ,  5.2 × 10⁻⁶ ,  1.1 × 10⁻⁵ ,  4.9 × 10⁻⁵

Check. As ordinary numbers: 0.0000008, 0.0000052, 0.000011, 0.000049 — these increase along the list. ✓

Watch out. A negative power is smaller than a positive power, and a larger negative power (like −8) is smaller than −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. Multiply the front numbers, add the powers of 10, then make sure the front number is between 1 and 10.

1. Write the multiplication with the number parts together and the powers of 10 together. Multiplication can be done in any order, so we can group the front numbers and group the powers of 10.
   (2 × 10³) × (3 × 10⁴) = (2 × 3) × (10³ × 10⁴)
2. Multiply the front numbers. Do this like an ordinary decimal multiplication.
   2 × 3 = 6
3. Multiply the powers of 10: add the indices. 10^a × 10^b = 10^a+b.
   10³ × 10⁴ = 10^{3 + 4} = 10⁷
4. Put the two parts together.
   6 × 10⁷
5. Write the answer.
   6 × 10⁷

Check. As ordinary numbers: 2000 × 30000 = 60000000, which is 6 × 10⁷. ✓

Watch out. Add the powers when you multiply (do not multiply them). Always check the front number at the end: 12 × 10⁷ is not in standard form.

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. Multiply the front numbers, add the powers of 10, then make sure the front number is between 1 and 10.

1. Write the multiplication with the number parts together and the powers of 10 together. Multiplication can be done in any order, so we can group the front numbers and group the powers of 10.
   (4 × 10⁵) × (2 × 10⁻²) = (4 × 2) × (10⁵ × 10⁻²)
2. Multiply the front numbers. Do this like an ordinary decimal multiplication.
   4 × 2 = 8
3. Multiply the powers of 10: add the indices. 10^a × 10^b = 10^a+b.
   10⁵ × 10⁻² = 10^{5 + (−2)} = 10³
4. Put the two parts together.
   8 × 10³
5. Write the answer.
   8 × 10³

Check. As ordinary numbers: 400000 × 0.02 = 8000, which is 8 × 10³. ✓

Watch out. Add the powers when you multiply (do not multiply them). Always check the front number at the end: 12 × 10⁷ is not in standard form.

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. Multiply the front numbers, add the powers of 10, then make sure the front number is between 1 and 10.

1. Write the multiplication with the number parts together and the powers of 10 together. Multiplication can be done in any order, so we can group the front numbers and group the powers of 10.
   (1.5 × 10⁶) × (4 × 10³) = (1.5 × 4) × (10⁶ × 10³)
2. Multiply the front numbers. Do this like an ordinary decimal multiplication.
   Ignore the decimal points: 15 × 4 = 60
   There are 1 decimal place altogether in 1.5 and 4, so the answer has 1 decimal place: 1.5 × 4 = 6
3. Multiply the powers of 10: add the indices. 10^a × 10^b = 10^a+b.
   10⁶ × 10³ = 10^{6 + 3} = 10⁹
4. Put the two parts together.
   6 × 10⁹
5. Write the answer.
   6 × 10⁹

Check. As ordinary numbers: 1500000 × 4000 = 6000000000, which is 6 × 10⁹. ✓

Watch out. Add the powers when you multiply (do not multiply them). Always check the front number at the end: 12 × 10⁷ is not in standard form.

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. Multiply the front numbers, add the powers of 10, then make sure the front number is between 1 and 10.

1. Write the multiplication with the number parts together and the powers of 10 together. Multiplication can be done in any order, so we can group the front numbers and group the powers of 10.
   (2.5 × 10⁻³) × (4 × 10⁻²) = (2.5 × 4) × (10⁻³ × 10⁻²)
2. Multiply the front numbers. Do this like an ordinary decimal multiplication.
   Ignore the decimal points: 25 × 4 = 100
   There are 1 decimal place altogether in 2.5 and 4, so the answer has 1 decimal place: 2.5 × 4 = 10
3. Multiply the powers of 10: add the indices. 10^a × 10^b = 10^a+b.
   10⁻³ × 10⁻² = 10^{−3 + (−2)} = 10⁻⁵
4. Put the two parts together.
   10 × 10⁻⁵
5. Check: is the front number between 1 and 10? 10 is 10 or more, so this is not standard form yet. Write 10 = 1 × 10 and combine the powers of 10.
   10 × 10⁻⁵ = 1 × 10 × 10⁻⁵ = 1 × 10^{-5 + 1} = 1 × 10⁻⁴
6. Write the answer.
   1 × 10⁻⁴

Check. As ordinary numbers: 0.0025 × 0.04 = 0.0001, which is 1 × 10⁻⁴. ✓

Watch out. Add the powers when you multiply (do not multiply them). Always check the front number at the end: 12 × 10⁷ is not in standard form.

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. Multiply the front numbers, add the powers of 10, then make sure the front number is between 1 and 10.

1. Write the multiplication with the number parts together and the powers of 10 together. Multiplication can be done in any order, so we can group the front numbers and group the powers of 10.
   (3 × 10⁸) × (3 × 10⁻⁵) = (3 × 3) × (10⁸ × 10⁻⁵)
2. Multiply the front numbers. Do this like an ordinary decimal multiplication.
   3 × 3 = 9
3. Multiply the powers of 10: add the indices. 10^a × 10^b = 10^a+b.
   10⁸ × 10⁻⁵ = 10^{8 + (−5)} = 10³
4. Put the two parts together.
   9 × 10³
5. Write the answer.
   9 × 10³

Check. As ordinary numbers: 300000000 × 0.00003 = 9000, which is 9 × 10³. ✓

Watch out. Add the powers when you multiply (do not multiply them). Always check the front number at the end: 12 × 10⁷ is not in standard form.

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. Divide the front numbers, subtract the powers of 10, then make sure the front number is between 1 and 10.

1. Write the division with the number parts together and the powers of 10 together. We can split the division into the front numbers and the powers of 10.
   (8 × 10⁹) ÷ (2 × 10⁴) = (8 ÷ 2) × (10⁹ ÷ 10⁴)
2. Divide the front numbers. Do this like an ordinary decimal division.
   8 ÷ 2 = 4
3. Divide the powers of 10: subtract the indices. 10^a ÷ 10^b = 10^a−b.
   10⁹ ÷ 10⁴ = 10^{9 − 4} = 10⁵
4. Put the two parts together.
   4 × 10⁵
5. Write the answer.
   4 × 10⁵

Check. Multiply back: 4 × 2 = 8 and the powers give 10⁵ × 10⁴ = 10⁹, so we get 8 × 10⁹ = 8000000000, which is the number we started with. ✓

Watch out. Subtract the powers when you divide, and take care with negative powers: 4 − (−2) = 6. If the front number comes out less than 1, adjust it.

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. Divide the front numbers, subtract the powers of 10, then make sure the front number is between 1 and 10.

1. Write the division with the number parts together and the powers of 10 together. We can split the division into the front numbers and the powers of 10.
   (6 × 10⁷) ÷ (3 × 10⁻²) = (6 ÷ 3) × (10⁷ ÷ 10⁻²)
2. Divide the front numbers. Do this like an ordinary decimal division.
   6 ÷ 3 = 2
3. Divide the powers of 10: subtract the indices. 10^a ÷ 10^b = 10^a−b.
   10⁷ ÷ 10⁻² = 10^{7 − (−2)} = 10⁹
4. Put the two parts together.
   2 × 10⁹
5. Write the answer.
   2 × 10⁹

Check. Multiply back: 2 × 3 = 6 and the powers give 10⁹ × 10⁻² = 10⁷, so we get 6 × 10⁷ = 60000000, which is the number we started with. ✓

Watch out. Subtract the powers when you divide, and take care with negative powers: 4 − (−2) = 6. If the front number comes out less than 1, adjust it.

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. Divide the front numbers, subtract the powers of 10, then make sure the front number is between 1 and 10.

1. Write the division with the number parts together and the powers of 10 together. We can split the division into the front numbers and the powers of 10.
   (9 × 10⁵) ÷ (4 × 10²) = (9 ÷ 4) × (10⁵ ÷ 10²)
2. Divide the front numbers. Do this like an ordinary decimal division.
   9 ÷ 4 = 2.25
3. Divide the powers of 10: subtract the indices. 10^a ÷ 10^b = 10^a−b.
   10⁵ ÷ 10² = 10^{5 − 2} = 10³
4. Put the two parts together.
   2.25 × 10³
5. Write the answer.
   2.25 × 10³

Check. Multiply back: 2.25 × 4 = 9 and the powers give 10³ × 10² = 10⁵, so we get 9 × 10⁵ = 900000, which is the number we started with. ✓

Watch out. Subtract the powers when you divide, and take care with negative powers: 4 − (−2) = 6. If the front number comes out less than 1, adjust it.

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. Divide the front numbers, subtract the powers of 10, then make sure the front number is between 1 and 10.

1. Write the division with the number parts together and the powers of 10 together. We can split the division into the front numbers and the powers of 10.
   (3 × 10⁵) ÷ (5 × 10⁻²) = (3 ÷ 5) × (10⁵ ÷ 10⁻²)
2. Divide the front numbers. Do this like an ordinary decimal division.
   3 ÷ 5 = 0.6
3. Divide the powers of 10: subtract the indices. 10^a ÷ 10^b = 10^a−b.
   10⁵ ÷ 10⁻² = 10^{5 − (−2)} = 10⁷
4. Put the two parts together.
   0.6 × 10⁷
5. Check: is the front number between 1 and 10? 0.6 is less than 1, so this is not standard form yet. Write 0.6 = 6 × 10⁻¹ and combine the powers of 10.
   0.6 × 10⁷ = 6 × 10⁻¹ × 10⁷ = 6 × 10^{−1 + 7} = 6 × 10⁶
6. Write the answer.
   6 × 10⁶

Check. Multiply back: 6 × 5 = 30 and the powers give 10⁶ × 10⁻² = 10⁴, so we get 30 × 10⁴ = 300000, which is the number we started with. ✓

Watch out. Subtract the powers when you divide, and take care with negative powers: 4 − (−2) = 6. If the front number comes out less than 1, adjust it.

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. Divide the front numbers, subtract the powers of 10, then make sure the front number is between 1 and 10.

1. Write the division with the number parts together and the powers of 10 together. We can split the division into the front numbers and the powers of 10.
   (4.5 × 10⁻³) ÷ (1.5 × 10²) = (4.5 ÷ 1.5) × (10⁻³ ÷ 10²)
2. Divide the front numbers. Do this like an ordinary decimal division.
   Multiply both numbers by 10 to make them whole (the answer does not change): 4.5 ÷ 1.5 = 45 ÷ 15 = 3
3. Divide the powers of 10: subtract the indices. 10^a ÷ 10^b = 10^a−b.
   10⁻³ ÷ 10² = 10^{−3 − 2} = 10⁻⁵
4. Put the two parts together.
   3 × 10⁻⁵
5. Write the answer.
   3 × 10⁻⁵

Check. Multiply back: 3 × 1.5 = 4.5 and the powers give 10⁻⁵ × 10² = 10⁻³, so we get 4.5 × 10⁻³ = 0.0045, which is the number we started with. ✓

Watch out. Subtract the powers when you divide, and take care with negative powers: 4 − (−2) = 6. If the front number comes out less than 1, adjust it.

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. Divide the front numbers, subtract the powers of 10, then make sure the front number is between 1 and 10.

1. Write the division with the number parts together and the powers of 10 together. We can split the division into the front numbers and the powers of 10.
   (1.2 × 10⁶) ÷ (4 × 10⁻²) = (1.2 ÷ 4) × (10⁶ ÷ 10⁻²)
2. Divide the front numbers. Do this like an ordinary decimal division.
   Multiply both numbers by 10 to make them whole (the answer does not change): 1.2 ÷ 4 = 12 ÷ 40 = 0.3
3. Divide the powers of 10: subtract the indices. 10^a ÷ 10^b = 10^a−b.
   10⁶ ÷ 10⁻² = 10^{6 − (−2)} = 10⁸
4. Put the two parts together.
   0.3 × 10⁸
5. Check: is the front number between 1 and 10? 0.3 is less than 1, so this is not standard form yet. Write 0.3 = 3 × 10⁻¹ and combine the powers of 10.
   0.3 × 10⁸ = 3 × 10⁻¹ × 10⁸ = 3 × 10^{−1 + 8} = 3 × 10⁷
6. Write the answer.
   3 × 10⁷

Check. Multiply back: 3 × 4 = 12 and the powers give 10⁷ × 10⁻² = 10⁵, so we get 12 × 10⁵ = 1200000, which is the number we started with. ✓

Watch out. Subtract the powers when you divide, and take care with negative powers: 4 − (−2) = 6. If the front number comes out less than 1, adjust it.

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. Add standard form numbers: give them the same power of 10, add the front numbers, then tidy the answer back into standard form.

1. Make the powers of 10 the same. We can only add when the powers of 10 match, just as we can only add fractions with the same denominator. Change the number with the smaller power so that it uses 10⁴. Each increase of 1 in the power means moving the decimal point of the front number one place to the left.
   8.5 × 10³ = 0.85 × 10⁴
2. Add the front numbers. The power of 10 stays the same: it just tells us the size of the unit.
   3.2 + 0.85 = 4.05
   so the result is 4.05 × 10⁴
3. Write the answer.
   4.05 × 10⁴

Check. As ordinary numbers: 32000 + 8500 = 40500, which is 4.05 × 10⁴. ✓

Watch out. You cannot just add the front numbers when the powers are different: 3 × 10⁴ + 2 × 10³ is not 5 × 10⁷.

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. Add standard form numbers: give them the same power of 10, add the front numbers, then tidy the answer back into standard form.

1. Look at the powers of 10. Both numbers have the same power, 10⁶, so the pieces are the same size and we can add the front numbers straight away. (It is like adding 3 lots of something and 5 lots of the same thing.)
   Both numbers are in units of 10⁶.
2. Add the front numbers. The power of 10 stays the same: it just tells us the size of the unit.
   5.2 + 4.1 = 9.3
   so the result is 9.3 × 10⁶
3. Write the answer.
   9.3 × 10⁶

Check. As ordinary numbers: 5200000 + 4100000 = 9300000, which is 9.3 × 10⁶. ✓

Watch out. You cannot just add the front numbers when the powers are different: 3 × 10⁴ + 2 × 10³ is not 5 × 10⁷.

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. Subtract standard form numbers: give them the same power of 10, subtract the front numbers, then tidy the answer back into standard form.

1. Make the powers of 10 the same. We can only subtract when the powers of 10 match, just as we can only subtract fractions with the same denominator. Change the number with the smaller power so that it uses 10⁵. Each increase of 1 in the power means moving the decimal point of the front number one place to the left.
   2.5 × 10⁴ = 0.25 × 10⁵
2. Subtract the front numbers. The power of 10 stays the same: it just tells us the size of the unit.
   6 − 0.25 = 5.75
   so the result is 5.75 × 10⁵
3. Write the answer.
   5.75 × 10⁵

Check. As ordinary numbers: 600000 − 25000 = 575000, which is 5.75 × 10⁵. ✓

Watch out. You cannot just add the front numbers when the powers are different: 3 × 10⁴ + 2 × 10³ is not 5 × 10⁷.

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. Subtract standard form numbers: give them the same power of 10, subtract the front numbers, then tidy the answer back into standard form.

1. Look at the powers of 10. Both numbers have the same power, 10⁻³, so the pieces are the same size and we can subtract the front numbers straight away. (It is like subtracting 3 lots of something and 5 lots of the same thing.)
   Both numbers are in units of 10⁻³.
2. Subtract the front numbers. The power of 10 stays the same: it just tells us the size of the unit.
   7.2 − 3.1 = 4.1
   so the result is 4.1 × 10⁻³
3. Write the answer.
   4.1 × 10⁻³

Check. As ordinary numbers: 0.0072 − 0.0031 = 0.0041, which is 4.1 × 10⁻³. ✓

Watch out. You cannot just add the front numbers when the powers are different: 3 × 10⁴ + 2 × 10³ is not 5 × 10⁷.

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. Add standard form numbers: give them the same power of 10, add the front numbers, then tidy the answer back into standard form.

1. Make the powers of 10 the same. We can only add when the powers of 10 match, just as we can only add fractions with the same denominator. Change the number with the smaller power so that it uses 10⁷. Each increase of 1 in the power means moving the decimal point of the front number one place to the left.
   7.5 × 10⁶ = 0.75 × 10⁷
2. Add the front numbers. The power of 10 stays the same: it just tells us the size of the unit.
   9.5 + 0.75 = 10.25
   so the result is 10.25 × 10⁷
3. Check: is the front number between 1 and 10? 10.25 is 10 or more, so write it with one non-zero digit before the decimal point and adjust the power.
   10.25 × 10⁷ = 1.025 × 10¹ × 10⁷ = 1.025 × 10⁸
4. Write the answer.
   1.025 × 10⁸

Check. As ordinary numbers: 95000000 + 7500000 = 102500000, which is 1.025 × 10⁸. ✓

Watch out. You cannot just add the front numbers when the powers are different: 3 × 10⁴ + 2 × 10³ is not 5 × 10⁷.

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. Subtract standard form numbers: give them the same power of 10, subtract the front numbers, then tidy the answer back into standard form.

1. Look at the powers of 10. Both numbers have the same power, 10⁴, so the pieces are the same size and we can subtract the front numbers straight away. (It is like subtracting 3 lots of something and 5 lots of the same thing.)
   Both numbers are in units of 10⁴.
2. Subtract the front numbers. The power of 10 stays the same: it just tells us the size of the unit.
   5.2 − 4.9 = 0.3
   so the result is 0.3 × 10⁴
3. Check: is the front number between 1 and 10? 0.3 is less than 1, so write it with one non-zero digit before the decimal point and adjust the power.
   0.3 × 10⁴ = 3 × 10⁻¹ × 10⁴ = 3 × 10³
4. Write the answer.
   3 × 10³

Check. As ordinary numbers: 52000 − 49000 = 3000, which is 3 × 10³. ✓

Watch out. You cannot just add the front numbers when the powers are different: 3 × 10⁴ + 2 × 10³ is not 5 × 10⁷.

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. Raise the front number to the power, multiply the index of 10 by the power, then tidy up.

1. Write the power of a product as separate powers. Both the front number and the power of 10 are raised to the power 3: (A × 10ⁿ)³ = A³ × (10ⁿ)³.
   (2 × 10³)³ = 2³ × (10³)³
2. Work out 2³. Multiply 2 by itself 3 times.
   2³ = 2 × 2 × 2 = 8
3. Work out the power of 10: multiply the indices. (10^a)^b = 10^a×b.
   (10³)³ = 10^{3 × 3} = 10⁹
4. Put the parts together.
   8 × 10⁹
5. Check the front number. It is between 1 and 10, so this is already in standard form.
   8 × 10⁹

Check. As ordinary numbers: 2000³ = 8000000000, which is 8 × 10⁹. ✓

Watch out. The power applies to both parts. (2 × 10³)² is 4 × 10⁶, not 2 × 10⁶.

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. Raise the front number to the power, multiply the index of 10 by the power, then tidy up.

1. Write the power of a product as separate powers. Both the front number and the power of 10 are raised to the power 2: (A × 10ⁿ)² = A² × (10ⁿ)².
   (4 × 10⁻³)² = 4² × (10⁻³)²
2. Work out 4². Multiply 4 by itself 2 times.
   4² = 4 × 4 = 16
3. Work out the power of 10: multiply the indices. (10^a)^b = 10^a×b.
   (10⁻³)² = 10^{−3 × 2} = 10⁻⁶
4. Put the parts together.
   16 × 10⁻⁶
5. Tidy into standard form. 16 is 10 or more, so write 16 = 1.6 × 10¹.
   16 × 10⁻⁶ = 1.6 × 10¹ × 10⁻⁶ = 1.6 × 10⁻⁵

Check. As ordinary numbers: 0.004² = 0.000016, which is 1.6 × 10⁻⁵. ✓

Watch out. The power applies to both parts. (2 × 10³)² is 4 × 10⁶, not 2 × 10⁶.

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. Raise the front number to the power, multiply the index of 10 by the power, then tidy up.

1. Write the power of a product as separate powers. Both the front number and the power of 10 are raised to the power 2: (A × 10ⁿ)² = A² × (10ⁿ)².
   (5 × 10⁴)² = 5² × (10⁴)²
2. Work out 5². Multiply 5 by itself 2 times.
   5² = 5 × 5 = 25
3. Work out the power of 10: multiply the indices. (10^a)^b = 10^a×b.
   (10⁴)² = 10^{4 × 2} = 10⁸
4. Put the parts together.
   25 × 10⁸
5. Tidy into standard form. 25 is 10 or more, so write 25 = 2.5 × 10¹.
   25 × 10⁸ = 2.5 × 10¹ × 10⁸ = 2.5 × 10⁹

Check. As ordinary numbers: 50000² = 2500000000, which is 2.5 × 10⁹. ✓

Watch out. The power applies to both parts. (2 × 10³)² is 4 × 10⁶, not 2 × 10⁶.

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. Make the power of 10 even, take the square root of the front number, and halve the power.

1. Take the square root of each part. √(a × b) = √a × √b.
   √9 = 3 (because 3 × 3 = 9)   and   √10⁶ = 10^{6 ÷ 2} = 10³
2. Put the parts together.
   3 × 10³

Check. Square the answer: (3 × 10³)² = 9 × 10⁶, which is 9 × 10⁶ after tidying. ✓

Watch out. Halve the power of 10: √(10⁶) = 10³, not 10⁶ ÷ 2 = 5 × 10⁵.

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. Make the power of 10 even, take the square root of the front number, and halve the power.

1. Make the power of 10 even. We need to halve the index, and an odd index cannot be halved to a whole number. Move one factor of 10 into the front number.
   6.4 × 10⁷ = 64 × 10⁶
2. Take the square root of each part. √(a × b) = √a × √b.
   √64 = 8 (because 8 × 8 = 64)   and   √10⁶ = 10^{6 ÷ 2} = 10³
3. Put the parts together.
   8 × 10³

Check. Square the answer: (8 × 10³)² = 64 × 10⁶, which is 6.4 × 10⁷ after tidying. ✓

Watch out. Halve the power of 10: √(10⁶) = 10³, not 10⁶ ÷ 2 = 5 × 10⁵.

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. Make the power of 10 even, take the square root of the front number, and halve the power.

1. Make the power of 10 even. We need to halve the index, and an odd index cannot be halved to a whole number. Move one factor of 10 into the front number.
   2.5 × 10⁻⁵ = 25 × 10⁻⁶
2. Take the square root of each part. √(a × b) = √a × √b.
   √25 = 5 (because 5 × 5 = 25)   and   √10⁻⁶ = 10^{−6 ÷ 2} = 10⁻³
3. Put the parts together.
   5 × 10⁻³

Check. Square the answer: (5 × 10⁻³)² = 25 × 10⁻⁶, which is 2.5 × 10⁻⁵ after tidying. ✓

Watch out. Halve the power of 10: √(10⁶) = 10³, not 10⁶ ÷ 2 = 5 × 10⁵.

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. Make the power of 10 even, take the square root of the front number, and halve the power.

1. Make the power of 10 even. We need to halve the index, and an odd index cannot be halved to a whole number. Move one factor of 10 into the front number.
   1.6 × 10⁻³ = 16 × 10⁻⁴
2. Take the square root of each part. √(a × b) = √a × √b.
   √16 = 4 (because 4 × 4 = 16)   and   √10⁻⁴ = 10^{−4 ÷ 2} = 10⁻²
3. Put the parts together.
   4 × 10⁻²

Check. Square the answer: (4 × 10⁻²)² = 16 × 10⁻⁴, which is 1.6 × 10⁻³ after tidying. ✓

Watch out. Halve the power of 10: √(10⁶) = 10³, not 10⁶ ÷ 2 = 5 × 10⁵.

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. Use the ×10^x key on a calculator, then write the display in standard form to the accuracy asked for.

1. Enter the calculation using the ×10^x key. The ×10^x key (on some calculators it is EXP) types "× 10 to the power" in one go. For a negative power, press the (−) key before the power number. Brackets stop the calculator mixing the numbers up.
   Press: 4.5 [×10^x] 7 [×] 2.4 [×10^x] [(−)] 3 [=]
2. Read the answer from the display. Your calculator may show the answer as an ordinary number or in its own standard form style, such as 2.4E−3. The digits are what matter.
   Display: 1.08 × 10⁵ (about)
3. Round to 3 significant figures and write in standard form. Round the front number A to 3 significant figures: look at the digit after the last one you keep, and round up if it is 5 or more. The power of 10 stays the same unless the rounding makes A reach 10.
   1.08 × 10⁵

Check. Estimate by rounding each front number to the nearest whole number: (5 × 10⁷) × (2 × 10⁻³) ≈ 1 × 10⁵, which is close to our answer. ✓

Watch out. Use the ×10^x key, not separate × 10 ^ keys, unless you use brackets. Always check the power is sensible with a quick estimate.

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. Use the ×10^x key on a calculator, then write the display in standard form to the accuracy asked for.

1. Enter the calculation using the ×10^x key. The ×10^x key (on some calculators it is EXP) types "× 10 to the power" in one go. For a negative power, press the (−) key before the power number. Brackets stop the calculator mixing the numbers up.
   Press: ( 5.97 [×10^x] 24 ) [÷] ( 7.35 [×10^x] 22 ) [=]
2. Read the answer from the display. Your calculator may show the answer as an ordinary number or in its own standard form style, such as 2.4E−3. The digits are what matter.
   Display: 8.12244898 × 10¹ (about)
3. Round to 3 significant figures and write in standard form. Round the front number A to 3 significant figures: look at the digit after the last one you keep, and round up if it is 5 or more. The power of 10 stays the same unless the rounding makes A reach 10.
   8.12 × 10¹

Check. Estimate by rounding each front number to the nearest whole number: (6 × 10²⁴) ÷ (7 × 10²²) ≈ 9 × 10¹, which is close to our answer. ✓

Watch out. Use the ×10^x key, not separate × 10 ^ keys, unless you use brackets. Always check the power is sensible with a quick estimate.

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. Use the ×10^x key on a calculator, then write the display in standard form to the accuracy asked for.

1. Enter the calculation using the ×10^x key. The ×10^x key (on some calculators it is EXP) types "× 10 to the power" in one go. For a negative power, press the (−) key before the power number. Brackets stop the calculator mixing the numbers up.
   Press: ( 3.2 [×10^x] 5 ) [×] ( 4.7 [×10^x] [(−)] 2 ) [÷] ( 1.6 [×10^x] 4 ) [=]
2. Read the answer from the display. Your calculator may show the answer as an ordinary number or in its own standard form style, such as 2.4E−3. The digits are what matter.
   Display: 9.4 × 10⁻¹ (about)
3. Round to 3 significant figures and write in standard form. Round the front number A to 3 significant figures: look at the digit after the last one you keep, and round up if it is 5 or more. The power of 10 stays the same unless the rounding makes A reach 10.
   9.40 × 10⁻¹

Check. Estimate by rounding each front number to the nearest whole number: (3 × 10⁵) × (5 × 10⁻²) ÷ (2 × 10⁴) ≈ 8 × 10⁻¹, which is close to our answer. ✓

Watch out. Use the ×10^x key, not separate × 10 ^ keys, unless you use brackets. Always check the power is sensible with a quick estimate.

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. Use the ×10^x key on a calculator, then write the display in standard form to the accuracy asked for.

1. Enter the calculation using the ×10^x key. The ×10^x key (on some calculators it is EXP) types "× 10 to the power" in one go. For a negative power, press the (−) key before the power number. Brackets stop the calculator mixing the numbers up.
   Press: 6.84 [×10^x] 8 [+] 3.9 [×10^x] 9 [=]
2. Read the answer from the display. Your calculator may show the answer as an ordinary number or in its own standard form style, such as 2.4E−3. The digits are what matter.
   Display: 4.584 × 10⁹ (about)
3. Round to 3 significant figures and write in standard form. Round the front number A to 3 significant figures: look at the digit after the last one you keep, and round up if it is 5 or more. The power of 10 stays the same unless the rounding makes A reach 10.
   4.58 × 10⁹

Check. Estimate by rounding each front number to the nearest whole number: (7 × 10⁸) + (4 × 10⁹) ≈ 5 × 10⁹, which is close to our answer. ✓

Watch out. Use the ×10^x key, not separate × 10 ^ keys, unless you use brackets. Always check the power is sensible with a quick estimate.

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. Use the ×10^x key on a calculator, then write the display in standard form to the accuracy asked for.

1. Enter the calculation using the ×10^x key. The ×10^x key (on some calculators it is EXP) types "× 10 to the power" in one go. For a negative power, press the (−) key before the power number. Brackets stop the calculator mixing the numbers up.
   Press: ( 2.7 [×10^x] 4 ) [x²] [=]
2. Read the answer from the display. Your calculator may show the answer as an ordinary number or in its own standard form style, such as 2.4E−3. The digits are what matter.
   Display: 7.29 × 10⁸ (about)
3. Round to 3 significant figures and write in standard form. Round the front number A to 3 significant figures: look at the digit after the last one you keep, and round up if it is 5 or more. The power of 10 stays the same unless the rounding makes A reach 10.
   7.29 × 10⁸

Check. Estimate by rounding each front number to the nearest whole number: (3 × 10⁴)² ≈ 9 × 10⁸, which is close to our answer. ✓

Watch out. Use the ×10^x key, not separate × 10 ^ keys, unless you use brackets. Always check the power is sensible with a quick estimate.

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. Use the ×10^x key on a calculator, then write the display in standard form to the accuracy asked for.

1. Enter the calculation using the ×10^x key. The ×10^x key (on some calculators it is EXP) types "× 10 to the power" in one go. For a negative power, press the (−) key before the power number. Brackets stop the calculator mixing the numbers up.
   Press: [√] ( 8.1 [×10^x] 7 ) [=]
2. Read the answer from the display. Your calculator may show the answer as an ordinary number or in its own standard form style, such as 2.4E−3. The digits are what matter.
   Display: 9 × 10³ (about)
3. Round to 3 significant figures and write in standard form. Round the front number A to 3 significant figures: look at the digit after the last one you keep, and round up if it is 5 or more. The power of 10 stays the same unless the rounding makes A reach 10.
   9.00 × 10³

Check. Estimate by rounding each front number to the nearest whole number: √(8 × 10⁷) ≈ 9 × 10³, which is close to our answer. ✓

Watch out. Use the ×10^x key, not separate × 10 ^ keys, unless you use brackets. Always check the power is sensible with a quick estimate.

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. Use the ×10^x key on a calculator, then write the display in standard form to the accuracy asked for.

1. Enter the calculation using the ×10^x key. The ×10^x key (on some calculators it is EXP) types "× 10 to the power" in one go. For a negative power, press the (−) key before the power number. Brackets stop the calculator mixing the numbers up.
   Press: 9.2 [×10^x] [(−)] 3 [−] 3.7 [×10^x] [(−)] 4 [=]
2. Read the answer from the display. Your calculator may show the answer as an ordinary number or in its own standard form style, such as 2.4E−3. The digits are what matter.
   Display: 8.83 × 10⁻³ (about)
3. Round to 3 significant figures and write in standard form. Round the front number A to 3 significant figures: look at the digit after the last one you keep, and round up if it is 5 or more. The power of 10 stays the same unless the rounding makes A reach 10.
   8.83 × 10⁻³

Check. Estimate by rounding each front number to the nearest whole number: (9 × 10⁻³) − (4 × 10⁻⁴) ≈ 9 × 10⁻³, which is close to our answer. ✓

Watch out. Use the ×10^x key, not separate × 10 ^ keys, unless you use brackets. Always check the power is sensible with a quick estimate.