Indices and standard form

Method. 3⁴ means 4 copies of 3 multiplied together.

1. Write the power as a multiplication. The index 4 tells us how many times the base 3 is used.
   3 × 3 × 3 × 3
2. Multiply one step at a time. Work from left to right. Each answer is the start of the next step.
   3, 9, 27, 81
3. Write the answer. This is the value of the power.
   81

Check. Count the copies of 3: there are 4. The answer 81 is right.

Watch out. Do not multiply the base by the index. 3⁴ is 3 × 3 × ... with 4 copies, not 3 × 4.

Method. 2⁶ means 6 copies of 2 multiplied together.

1. Write the power as a multiplication. The index 6 tells us how many times the base 2 is used.
   2 × 2 × 2 × 2 × 2 × 2
2. Multiply one step at a time. Work from left to right. Each answer is the start of the next step.
   2, 4, 8, 16, 32, 64
3. Write the answer. This is the value of the power.
   64

Check. Count the copies of 2: there are 6. The answer 64 is right.

Watch out. Do not multiply the base by the index. 2⁶ is 2 × 2 × ... with 6 copies, not 2 × 6.

Method. 5³ means 3 copies of 5 multiplied together.

1. Write the power as a multiplication. The index 3 tells us how many times the base 5 is used.
   5 × 5 × 5
2. Multiply one step at a time. Work from left to right. Each answer is the start of the next step.
   5, 25, 125
3. Write the answer. This is the value of the power.
   125

Check. Count the copies of 5: there are 3. The answer 125 is right.

Watch out. Do not multiply the base by the index. 5³ is 5 × 5 × ... with 3 copies, not 5 × 3.

Method. 10⁴ means 4 copies of 10 multiplied together.

1. Write the power as a multiplication. The index 4 tells us how many times the base 10 is used.
   10 × 10 × 10 × 10
2. Multiply one step at a time. Work from left to right. Each answer is the start of the next step.
   10, 100, 1,000, 10,000
3. Write the answer. This is the value of the power.
   10,000

Check. Count the copies of 10: there are 4. The answer 10,000 is right.

Watch out. Do not multiply the base by the index. 10⁴ is 10 × 10 × ... with 4 copies, not 10 × 4.

Method. 4³ means 3 copies of 4 multiplied together.

1. Write the power as a multiplication. The index 3 tells us how many times the base 4 is used.
   4 × 4 × 4
2. Multiply one step at a time. Work from left to right. Each answer is the start of the next step.
   4, 16, 64
3. Write the answer. This is the value of the power.
   64

Check. Count the copies of 4: there are 3. The answer 64 is right.

Watch out. Do not multiply the base by the index. 4³ is 4 × 4 × ... with 3 copies, not 4 × 3.

Method. 7² means 2 copies of 7 multiplied together.

1. Write the power as a multiplication. The index 2 tells us how many times the base 7 is used.
   7 × 7
2. Multiply one step at a time. Work from left to right. Each answer is the start of the next step.
   7, 49
3. Write the answer. This is the value of the power.
   49

Check. Count the copies of 7: there are 2. The answer 49 is right.

Watch out. Do not multiply the base by the index. 7² is 7 × 7 × ... with 2 copies, not 7 × 2.

Method. Count the copies of 6 and write the index.

1. Count the copies. There are five 6s.
   five copies of 6
2. Write the power. The base is 6 and the index is 5.
   6⁵

Check. 6⁵ = 7,776, and 6 × 6 × 6 × 6 × 6 is also 7,776.

Method. Count the copies of 9 and write the index.

1. Count the copies. There are four 9s.
   four copies of 9
2. Write the power. The base is 9 and the index is 4.
   9⁴

Check. 9⁴ = 6,561.

Method. Work out both powers, then compare the answers.

1. Work out 2⁵. Five 2s multiplied together.
   2 × 2 × 2 × 2 × 2 = 32
2. Work out 5². Two 5s multiplied together.
   5 × 5 = 25
3. Compare. 32 is bigger than 25.
   32 > 25
4. Write the answer.
   2⁵ is bigger

Check. 2⁵ = 32 and 5² = 25, so 2⁵ is the bigger number.

Watch out. Do not swap the base and the index. 5² is 25, not 32.

Method. Work out each power and compare.

1. Work out 3². Two 3s multiplied together.
   3 × 3 = 9
2. Work out 2³. Three 2s multiplied together.
   2 × 2 × 2 = 8
3. Compare. 9 and 8 are different.
   9 ≠ 8
4. Write the answer.
   No

Check. 3² = 9 and 2³ = 8.

Watch out. Swapping the base and index usually gives a different answer.

Method. A cube number is a number multiplied by itself three times.

1. Write the power in full. Three 5s multiplied together.
   5 × 5 × 5
2. Multiply the first two. 5 × 5 = 25.
   5 × 5 = 25
3. Multiply by the third 5. 25 × 5 = 125.
   25 × 5 = 125

Check. 125 is 5 cubed, so it is a cube number.

Watch out. A cube number is a power of 3, not a power of 2.

Method. Use the hint: 2¹⁰ is two copies of 2⁵ multiplied together.

1. Write 2¹⁰ as two 2⁵. Ten 2s can be split into two groups of five.
   2¹⁰ = 2⁵ × 2⁵
2. Use the value of 2⁵. We know 2⁵ = 32.
   32 × 32
3. Multiply. 32 × 32 = 1,024.
   1,024

Check. 1,024 is the value of 2 doubled ten times.

Method. Work out both powers and compare them.

1. Work out 4². Two 4s multiplied together.
   4 × 4 = 16
2. Work out 2⁴. Four 2s multiplied together.
   2 × 2 × 2 × 2 = 16
3. Compare. Both are 16, so these two powers are equal.
   16 = 16
4. Find another pair. Try 3² = 9 and 2³ = 8. These are different.
   9 ≠ 8
5. Write the answer.
   16

Check. Two different powers can give the same value, so always work each one out before you compare.

Watch out. Swapping the base and the index does not always change the answer, but it often does.

Method. Multiply powers with the same base by adding the indices.

1. Check the bases are the same. The index laws only work when the base is the same.
   Both bases are 2.
2. Add the indices. When powers with the same base are multiplied, add the indices.
   3 + 5 = 8
3. Check with numbers. Work out both powers and multiply them.
   8 × 32 = 256
4. Write the answer.
   2⁸

Check. 8 × 32 = 256, and 2⁸ is also 256.

Watch out. Add the indices. Do not multiply them, and do not multiply the bases.

Method. Multiply powers with the same base by adding the indices.

1. Check the bases are the same. The index laws only work when the base is the same.
   Both bases are 5.
2. Add the indices. When powers with the same base are multiplied, add the indices.
   2 + 1 = 3
3. Check with numbers. Work out both powers and multiply them.
   25 × 5 = 125
4. Write the answer.
   5³

Check. 25 × 5 = 125, and 5³ is also 125.

Watch out. Add the indices. Do not multiply them, and do not multiply the bases.

Method. Multiply powers with the same base by adding the indices.

1. Check the bases are the same. The index laws only work when the base is the same.
   Both bases are x.
2. Add the indices. When powers with the same base are multiplied, add the indices.
   5 + 3 = 8
3. Write the answer.
   x⁸

Check. x appears 8 times altogether, so the index is 8.

Watch out. Add the indices. Do not multiply them, and do not multiply the bases.

Method. Multiply powers with the same base by adding the indices.

1. Check the bases are the same. The index laws only work when the base is the same.
   Both bases are a.
2. Add the indices. When powers with the same base are multiplied, add the indices.
   4 + 6 = 10
3. Write the answer.
   a¹⁰

Check. a appears 10 times altogether, so the index is 10.

Watch out. Add the indices. Do not multiply them, and do not multiply the bases.

Method. Multiply powers with the same base by adding the indices.

1. Check the bases are the same. The index laws only work when the base is the same.
   Both bases are y.
2. Add the indices. When powers with the same base are multiplied, add the indices.
   2 + 7 = 9
3. Write the answer.
   y⁹

Check. y appears 9 times altogether, so the index is 9.

Watch out. Add the indices. Do not multiply them, and do not multiply the bases.

Method. Divide powers with the same base by subtracting the indices.

1. Check the bases are the same. The index laws need the same base.
   Both bases are 2.
2. Subtract the indices. When powers with the same base are divided, subtract the bottom index from the top index.
   9 − 4 = 5
3. Check with numbers. Multiply the answer by the bottom power. You should get the top power.
   32 × 16 = 512
4. Write the answer.
   2⁵

Check. 2⁵ × 2⁴ gives 2⁹, so the answer is right.

Watch out. Subtract the bottom index from the top index. Keep the order: top minus bottom.

Method. Divide powers with the same base by subtracting the indices.

1. Check the bases are the same. The index laws need the same base.
   Both bases are 7.
2. Subtract the indices. When powers with the same base are divided, subtract the bottom index from the top index.
   6 − 2 = 4
3. Check with numbers. Multiply the answer by the bottom power. You should get the top power.
   2,401 × 49 = 117,649
4. Write the answer.
   7⁴

Check. 7⁴ × 7² gives 7⁶, so the answer is right.

Watch out. Subtract the bottom index from the top index. Keep the order: top minus bottom.

Method. Divide powers with the same base by subtracting the indices.

1. Check the bases are the same. The index laws need the same base.
   Both bases are y.
2. Subtract the indices. When powers with the same base are divided, subtract the bottom index from the top index.
   9 − 4 = 5
3. Write the answer.
   y⁵

Check. y⁵ × y⁴ gives y⁹, so the answer is right.

Watch out. Subtract the bottom index from the top index. Keep the order: top minus bottom.

Method. Do the multiplication first, then the division. Both use the same base m.

1. Multiply the first two powers. Add the indices: 3 + 4 = 7.
   m³ × m⁴ = m⁷
2. Divide by the third power. Subtract the indices: 7 − 2 = 5.
   m⁷ ÷ m² = m⁵
3. Write the answer.
   m⁵

Check. Count the m factors: 3 + 4 − 2 = 5, so the index is 5.

Watch out. Work from left to right, and use one index law at a time.

Method. A power raised to another power: multiply the indices.

1. Look at the brackets. The outside index multiplies every index inside the brackets.
   (x³)²
2. Multiply the indices. The outside index is 2, so multiply 3 by 2.
   3 × 2 = 6
3. Write the answer.
   x⁶

Check. Write it out: x³ is repeated 2 times, which is 6 copies of x in total.

Watch out. Multiply the indices. Do not add them.

Method. Multiply the indices, then work out the number.

1. Multiply the indices. The outside index 3 multiplies the index 2 inside the brackets.
   2 × 3 = 6
2. Write the power of 2.
   2⁶
3. Work out the value. Six 2s multiplied together.
   2 × 2 × 2 × 2 × 2 × 2 = 64
4. Write the answer.
   2⁶ = 64

Check. (2²)³ = 4³ = 64.

Method. Add the indices, then work out the power.

1. Add the indices. The index of 3⁰ is 0, so 5 + 0 = 5.
   3^{5 + 0} = 3⁵
2. Work out the value. Five 3s multiplied together.
   3 × 3 × 3 × 3 × 3 = 243
3. Write the answer.
   3⁵ = 243

Check. The 3⁰ factor is 1, so the answer is the same as 3⁵.

Method. Work inside the brackets first, then divide.

1. Multiply inside the brackets. Add the indices: 4 + 5 = 9.
   x⁴ × x⁵ = x⁹
2. Divide by x³. Subtract the indices: 9 − 3 = 6.
   x⁹ ÷ x³ = x⁶
3. Write the answer.
   x⁶

Check. Count the x factors: 4 + 5 − 3 = 6.

Method. Work out both sides and compare.

1. Use the index law on the left. The bases are both 2, so add the indices: 3 + 4 = 7.
   2³ × 2⁴ = 2⁷
2. Work out the value of 2⁷. Seven 2s multiplied together.
   2⁷ = 128
3. Work out the value of 4⁷. Seven 4s multiplied together gives a much bigger number.
   4⁷ = 16,384
4. Compare. 128 and 16,384 are different.
   Not equal

Check. The index law only works with the same base. Here 4 is not 2.

Watch out. Do not change the base. Only the indices are added.

Method. Any number that is not zero, raised to the power 0, is 1.

1. Divide a power by itself. Dividing a number by itself always gives 1.
   7³ ÷ 7³ = 1
2. Use the division index law. Subtract the indices: 3 − 3 = 0.
   7³ ÷ 7³ = 7⁰
3. Write the answer. So the power to the zero is 1.
   7⁰ = 1

Check. Both routes give 1, so 7⁰ = 1 for any base that is not zero.

Watch out. Zero as an index means 1, not 0.

Method. Any number that is not zero, raised to the power 0, is 1.

1. Divide a power by itself. Dividing a number by itself always gives 1.
   12³ ÷ 12³ = 1
2. Use the division index law. Subtract the indices: 3 − 3 = 0.
   12³ ÷ 12³ = 12⁰
3. Write the answer. So the power to the zero is 1.
   12⁰ = 1

Check. Both routes give 1, so 12⁰ = 1 for any base that is not zero.

Watch out. Zero as an index means 1, not 0.

Method. A negative index means one over the positive power.

1. Look at the negative index. The minus sign tells us to take the reciprocal (flip the fraction).
   2⁻¹
2. Use the rule. Write 1 over 2 to the positive power 1.
   1 ÷ 2¹
3. Work out the power and write the answer. 2¹ is 2.
   1/2

Check. 1/2 is a small positive number, so a negative index does not give a negative answer.

Watch out. A negative index makes a small number. It does not make the answer negative.

Method. A negative index means one over the positive power.

1. Look at the negative index. The minus sign tells us to take the reciprocal (flip the fraction).
   2⁻³
2. Use the rule. Write 1 over 2 to the positive power 3.
   1 ÷ 2³
3. Work out the power and write the answer. 2³ is 8.
   1/8

Check. 1/8 is a small positive number, so a negative index does not give a negative answer.

Watch out. A negative index makes a small number. It does not make the answer negative.

Method. A negative index means one over the positive power.

1. Look at the negative index. The minus sign tells us to take the reciprocal (flip the fraction).
   10⁻²
2. Use the rule. Write 1 over 10 to the positive power 2.
   1 ÷ 10²
3. Work out the power and write the answer. 10² is 100.
   1/100

Check. 1/100 is a small positive number, so a negative index does not give a negative answer.

Watch out. A negative index makes a small number. It does not make the answer negative.

Method. A fraction with 1 on top is a negative index. Flip the power to the top: 1 over 2⁴ is 2⁻⁴.

1. Find the power in the bottom of the fraction. The bottom is 2 to the power 4.
   1/16 = 1/2⁴
2. Move the power to the top. Flipping the power to the top gives a negative index.
   1/2⁴ = 2⁻⁴
3. Write the answer.
   2⁻⁴

Check. 2⁻⁴ = 1/16, which is the fraction we started with.

Watch out. The base in the bottom of the fraction is the base of the index.

Method. A fraction with 1 on top is a negative index. Flip the power to the top: 1 over 7³ is 7⁻³.

1. Find the power in the bottom of the fraction. The bottom is 7 to the power 3.
   1/343 = 1/7³
2. Move the power to the top. Flipping the power to the top gives a negative index.
   1/7³ = 7⁻³
3. Write the answer.
   7⁻³

Check. 7⁻³ = 1/343, which is the fraction we started with.

Watch out. The base in the bottom of the fraction is the base of the index.

Method. A fraction with 1 on top is a negative index. Flip the power to the top: 1 over 3² is 3⁻².

1. Find the power in the bottom of the fraction. The bottom is 3 to the power 2.
   1/9 = 1/3²
2. Move the power to the top. Flipping the power to the top gives a negative index.
   1/3² = 3⁻²
3. Write the answer.
   3⁻²

Check. 3⁻² = 1/9, which is the fraction we started with.

Watch out. The base in the bottom of the fraction is the base of the index.

Method. Work out both values, then compare.

1. Work out 3⁰. Any number (not zero) to the power 0 is 1.
   3⁰ = 1
2. Work out 3⁻¹. A negative index gives one over the power.
   3⁻¹ = 1/3
3. Compare. 1 is bigger than 1/3.
   1 > 1/3
4. Write the answer.
   3⁰

Check. 1/3 is about 0.33, which is less than 1.

Method. Write the negative index as one over the power, then convert the fraction to a decimal.

1. Use the rule. A negative index means one over the positive power.
   10⁻³ = 1/10³
2. Work out the power. 10³ = 1,000.
   1/1,000
3. Write as a decimal. One thousandth is 0.001.
   1/1,000 = 0.001
4. Write the answer. The statement is true.
   True

Check. 0.001 has three decimal places, which matches the power 3.

Method. Use the rule for a negative index, then decide the sign.

1. Use the rule. A negative index means one over the positive power.
   4⁻² = 1/4²
2. Work out the power. 4² = 16.
   1/16
3. Decide the sign. 1/16 is a positive fraction.
   1/16 > 0
4. Write the answer.
   No

Check. A negative index gives a small positive fraction.

Method. Subtract the indices, then write the answer with a negative index.

1. Subtract the indices. The top index is 2 and the bottom index is 6, so 2 − 6 = −4.
   x^{2 − 6} = x⁻⁴
2. Write the answer. The power is negative, so the answer is written with a negative index.
   x⁻⁴

Check. x⁻⁴ = 1/x⁴, which is the same as x² ÷ x⁶ cancelled down.

Method. Add the indices, then use the zero rule.

1. Add the indices. Add 3 and −3: 3 + (−3) = 0.
   2^{3 + (−3)} = 2⁰
2. Use the zero rule. Any number (not zero) to the power 0 is 1.
   2⁰ = 1
3. Write the answer.
   1

Check. 2³ = 8 and 2⁻³ = 1/8, and 8 × 1/8 = 1.

Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places you moved.

1. Find the decimal point. For a whole number it is at the end. For a decimal it is after the units digit.
   45000
2. Move the decimal point 4 places left. The decimal point moves left for a large number, so the power is positive.
   4.5 with 4 moves
3. Write the answer in standard form. The power of 10 is 4.
   4.5 × 10⁴

Check. 4.5 × 10⁴ is 45,000, which is the number we started with.

Watch out. The power counts the moves. Moving left gives a positive power. Moving right gives a negative power.

Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places you moved.

1. Find the decimal point. For a whole number it is at the end. For a decimal it is after the units digit.
   3600000
2. Move the decimal point 6 places left. The decimal point moves left for a large number, so the power is positive.
   3.6 with 6 moves
3. Write the answer in standard form. The power of 10 is 6.
   3.6 × 10⁶

Check. 3.6 × 10⁶ is 3,600,000, which is the number we started with.

Watch out. The power counts the moves. Moving left gives a positive power. Moving right gives a negative power.

Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places you moved.

1. Find the decimal point. For a whole number it is at the end. For a decimal it is after the units digit.
   9300
2. Move the decimal point 3 places left. The decimal point moves left for a large number, so the power is positive.
   9.3 with 3 moves
3. Write the answer in standard form. The power of 10 is 3.
   9.3 × 10³

Check. 9.3 × 10³ is 9,300, which is the number we started with.

Watch out. The power counts the moves. Moving left gives a positive power. Moving right gives a negative power.

Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places you moved.

1. Find the decimal point. For a whole number it is at the end. For a decimal it is after the units digit.
   58000000000
2. Move the decimal point 10 places left. The decimal point moves left for a large number, so the power is positive.
   5.8 with 10 moves
3. Write the answer in standard form. The power of 10 is 10.
   5.8 × 10¹⁰

Check. 5.8 × 10¹⁰ is 58,000,000,000, which is the number we started with.

Watch out. The power counts the moves. Moving left gives a positive power. Moving right gives a negative power.

Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places you moved.

1. Find the decimal point. For a whole number it is at the end. For a decimal it is after the units digit.
   0.0042
2. Move the decimal point 3 places right. The decimal point moves right for a small number, so the power is negative.
   4.2 with 3 moves
3. Write the answer in standard form. The power of 10 is -3.
   4.2 × 10⁻³

Check. 4.2 × 10⁻³ is 0.0042, which is the number we started with.

Watch out. The power counts the moves. Moving left gives a positive power. Moving right gives a negative power.

Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places you moved.

1. Find the decimal point. For a whole number it is at the end. For a decimal it is after the units digit.
   0.0000051
2. Move the decimal point 6 places right. The decimal point moves right for a small number, so the power is negative.
   5.1 with 6 moves
3. Write the answer in standard form. The power of 10 is -6.
   5.1 × 10⁻⁶

Check. 5.1 × 10⁻⁶ is 0.0000051, which is the number we started with.

Watch out. The power counts the moves. Moving left gives a positive power. Moving right gives a negative power.

Method. Use the power of 10 to move the decimal point, then write the number in ordinary form.

1. Look at the power of 10. The power is 5.
   10⁵
2. Move the decimal point 5 places right. A positive power makes the number bigger, so the decimal point moves right.
   3.2 × 10⁵
3. Write the answer. Fill any gaps with zeros.
   320,000

Check. The answer should be bigger than 3.2, and 320,000 is.

Watch out. Fill any gaps with zeros. Count the places carefully.

Method. Use the power of 10 to move the decimal point, then write the number in ordinary form.

1. Look at the power of 10. The power is -4.
   10^-4
2. Move the decimal point 4 places left. A negative power makes the number smaller, so the decimal point moves left.
   6 × 10^-4
3. Write the answer. Fill any gaps with zeros.
   0.0006

Check. The answer should be smaller than 6, and 0.0006 is.

Watch out. Fill any gaps with zeros. Count the places carefully.

Method. Use the power of 10 to move the decimal point, then write the number in ordinary form.

1. Look at the power of 10. The power is -2.
   10^-2
2. Move the decimal point 2 places left. A negative power makes the number smaller, so the decimal point moves left.
   1.25 × 10^-2
3. Write the answer. Fill any gaps with zeros.
   0.0125

Check. The answer should be smaller than 1.25, and 0.0125 is.

Watch out. Fill any gaps with zeros. Count the places carefully.

Method. Use the power of 10 to move the decimal point, then write the number in ordinary form.

1. Look at the power of 10. The power is 8.
   10⁸
2. Move the decimal point 8 places right. A positive power makes the number bigger, so the decimal point moves right.
   1.5 × 10⁸
3. Write the answer. Fill any gaps with zeros.
   150,000,000

Check. The answer should be bigger than 1.5, and 150,000,000 is.

Watch out. Fill any gaps with zeros. Count the places carefully.

Method. Write both numbers in ordinary form, so they are easy to compare.

1. Write both numbers in ordinary form. Then the sizes are easy to see.
   3 × 10⁵ = 300,000 and 4.2 × 10⁴ = 42,000
2. Compare the sizes. The bigger number is the one with the bigger value.
   300,000 is the bigger number
3. Write the answer.
   3 × 10⁵

Check. A bigger power of 10 usually means a bigger number, but compare the numbers in front too when the powers match.

Watch out. Compare the whole numbers, not only the powers. A large front number can make a number bigger.

Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places you moved.

1. Find the decimal point. For a whole number it is at the end. For a decimal it is after the units digit.
   0.0000085
2. Move the decimal point 6 places right. The decimal point moves right for a small number, so the power is negative.
   8.5 with 6 moves
3. Write the answer in standard form. The power of 10 is -6.
   8.5 × 10⁻⁶

Check. 8.5 × 10⁻⁶ is 0.0000085, which is the number we started with.

Watch out. The power counts the moves. Moving left gives a positive power. Moving right gives a negative power.

Method. Move the decimal point so that there is one non-zero digit in front of it, then count the places.

1. Find the first non-zero digit. The 1 is the first digit that is not zero. Put the decimal point after it.
   0.00001 becomes 1 with the point moved
2. Count the places moved. The point moves five places to the right. The number is small, so the power is negative.
   5 places right
3. Write the answer. The power is −5.
   1 × 10⁻⁵

Check. 1 ÷ 10⁵ = 1/100,000 = 0.00001, which matches.

Method. Use the power of 10 to move the decimal point, then write the number in ordinary form.

1. Look at the power of 10. The power is -2.
   10^-2
2. Move the decimal point 2 places left. A negative power makes the number smaller, so the decimal point moves left.
   5 × 10^-2
3. Write the answer. Fill any gaps with zeros.
   0.05

Check. The answer should be smaller than 5, and 0.05 is.

Watch out. Fill any gaps with zeros. Count the places carefully.

Method. Multiply the front numbers, and add the powers of 10. Then check the front number is between 1 and 10.

1. Multiply the numbers in front. Multiply the two front numbers on their own.
   2 × 4 = 8
2. Add the powers of 10. When the base is 10 in both, add the indices.
   3 + 2 = 5
3. Write the answer in standard form.
   8 × 10⁵

Check. Ordinary numbers: 2,000 × 400 = 800,000.

Watch out. Do not add the front numbers. Only the powers of 10 are added.

Method. Multiply the front numbers, and add the powers of 10. Then check the front number is between 1 and 10.

1. Multiply the numbers in front. Multiply the two front numbers on their own.
   3 × 2 = 6
2. Add the powers of 10. When the base is 10 in both, add the indices.
   4 + 3 = 7
3. Write the answer in standard form.
   6 × 10⁷

Check. Ordinary numbers: 30,000 × 2,000 = 60,000,000.

Watch out. Do not add the front numbers. Only the powers of 10 are added.

Method. Multiply the front numbers, and add the powers of 10. Then check the front number is between 1 and 10.

1. Multiply the numbers in front. Multiply the two front numbers on their own.
   4 × 3 = 12
2. Add the powers of 10. When the base is 10 in both, add the indices.
   3 + 4 = 7
3. Make the front number between 1 and 10. It must be 1 or more and less than 10. Move the point and change the power.
   12 × 10⁷ = 1.2 × 10⁸
4. Write the answer in standard form.
   1.2 × 10⁸

Check. Ordinary numbers: 4,000 × 30,000 = 120,000,000.

Watch out. If the front number is 10 or more, divide it by 10 and add 1 to the power.

Method. Multiply the front numbers, and add the powers of 10. Then check the front number is between 1 and 10.

1. Multiply the numbers in front. Multiply the two front numbers on their own.
   2.5 × 4 = 10
2. Add the powers of 10. When the base is 10 in both, add the indices.
   3 + 2 = 5
3. Make the front number between 1 and 10. It must be 1 or more and less than 10. Move the point and change the power.
   10 × 10⁵ = 1 × 10⁶
4. Write the answer in standard form.
   1 × 10⁶

Check. Ordinary numbers: 2,500 × 400 = 1,000,000.

Watch out. If the front number is 10 or more, divide it by 10 and add 1 to the power.

Method. Divide the front numbers, and subtract the powers of 10. Then check the front number is between 1 and 10.

1. Divide the front numbers. Divide the first front number by the second.
   6 ÷ 2 = 3
2. Subtract the powers of 10. Subtract the bottom power from the top power.
   5 − 2 = 3
3. Write the answer in standard form.
   3 × 10³

Check. Ordinary numbers: 600,000 ÷ 200 = 3,000.

Watch out. Subtract the bottom power from the top power. Keep the order.

Method. Divide the front numbers, and subtract the powers of 10. Then check the front number is between 1 and 10.

1. Divide the front numbers. Divide the first front number by the second.
   8 ÷ 2 = 4
2. Subtract the powers of 10. Subtract the bottom power from the top power.
   6 − 2 = 4
3. Write the answer in standard form.
   4 × 10⁴

Check. Ordinary numbers: 8,000,000 ÷ 200 = 40,000.

Watch out. Subtract the bottom power from the top power. Keep the order.

Method. Divide the front numbers, and subtract the powers of 10. Then check the front number is between 1 and 10.

1. Divide the front numbers. Divide the first front number by the second.
   9 ÷ 3 = 3
2. Subtract the powers of 10. Subtract the bottom power from the top power.
   7 − 4 = 3
3. Write the answer in standard form.
   3 × 10³

Check. Ordinary numbers: 90,000,000 ÷ 30,000 = 3,000.

Watch out. Subtract the bottom power from the top power. Keep the order.

Method. Divide the front numbers, and subtract the powers of 10. Then check the front number is between 1 and 10.

1. Divide the front numbers. Divide the first front number by the second.
   1.2 ÷ 4 = 0.3
2. Subtract the powers of 10. Subtract the bottom power from the top power.
   8 − 3 = 5
3. Make the front number between 1 and 10. It must be 1 or more and less than 10.
   0.3 × 10⁵ = 3 × 10⁴
4. Write the answer in standard form.
   3 × 10⁴

Check. Ordinary numbers: 120,000,000 ÷ 4,000 = 30,000.

Watch out. Subtract the bottom power from the top power. Keep the order.

Method. Make the powers of 10 the same first. Then add the numbers in front.

1. Write both numbers with the same power of 10. Use the bigger power, 4, for both numbers.
   3 × 10⁴ and 5 × 10³ become 0.5 × 10⁴ instead
2. Add the numbers in front. The power of 10 stays the same when we add.
   3 + 0.5 = 3.5
3. Write the answer in standard form. Check the front number is between 1 and 10.
   3.5 × 10⁴

Check. Ordinary numbers: 30,000 + 5,000 = 35,000.

Watch out. Do not add the powers of 10 together. Make the powers the same first.

Method. Make the powers of 10 the same first. Then add the numbers in front.

1. Write both numbers with the same power of 10. Use the bigger power, 5, for both numbers.
   6 × 10⁵ and 2 × 10⁴ become 0.2 × 10⁵ instead
2. Add the numbers in front. The power of 10 stays the same when we add.
   6 + 0.2 = 6.2
3. Write the answer in standard form. Check the front number is between 1 and 10.
   6.2 × 10⁵

Check. Ordinary numbers: 600,000 + 20,000 = 620,000.

Watch out. Do not add the powers of 10 together. Make the powers the same first.

Method. Make the powers of 10 the same first. Then add the numbers in front.

1. Write both numbers with the same power of 10. Use the bigger power, 3, for both numbers.
   2 × 10³ and 5 × 10² become 0.5 × 10³ instead
2. Add the numbers in front. The power of 10 stays the same when we add.
   2 + 0.5 = 2.5
3. Write the answer in standard form. Check the front number is between 1 and 10.
   2.5 × 10³

Check. Ordinary numbers: 2,000 + 500 = 2,500.

Watch out. Do not add the powers of 10 together. Make the powers the same first.

Method. Multiply the front numbers, and add the powers of 10. Then check the front number is between 1 and 10.

1. Multiply the numbers in front. Multiply the two front numbers on their own.
   1.5 × 2 = 3
2. Add the powers of 10. When the base is 10 in both, add the indices.
   3 + 1 = 4
3. Write the answer in standard form.
   3 × 10⁴

Check. Ordinary numbers: 1,500 × 20 = 30,000.

Watch out. Do not add the front numbers. Only the powers of 10 are added.

Method. Multiply the front numbers, and add the powers of 10. Then check the front number is between 1 and 10.

1. Multiply the numbers in front. Multiply the two front numbers on their own.
   4 × 3 = 12
2. Add the powers of 10. When the base is 10 in both, add the indices.
   5 + -2 = 3
3. Make the front number between 1 and 10. It must be 1 or more and less than 10. Move the point and change the power.
   12 × 10³ = 1.2 × 10⁴
4. Write the answer in standard form.
   1.2 × 10⁴

Check. Ordinary numbers: 400,000 × 0.03 = 12,000.

Watch out. If the front number is 10 or more, divide it by 10 and add 1 to the power.

Method. Divide the front numbers, and subtract the powers of 10. Then check the front number is between 1 and 10.

1. Divide the front numbers. Divide the first front number by the second.
   9 ÷ 3 = 3
2. Subtract the powers of 10. Subtract the bottom power from the top power.
   6 − -2 = 8
3. Write the answer in standard form.
   3 × 10⁸

Check. Ordinary numbers: 9,000,000 ÷ 0.03 = 300,000,000.

Watch out. Subtract the bottom power from the top power. Keep the order.

Method. 5³ means 3 copies of 5 multiplied together.

1. Write the power as a multiplication. The index 3 tells us how many times the base 5 is used.
   5 × 5 × 5
2. Multiply one step at a time. Work from left to right. Each answer is the start of the next step.
   5, 25, 125
3. Write the answer. This is the value of the power.
   125

Check. Count the copies of 5: there are 3. The answer 125 is right.

Watch out. Do not multiply the base by the index. 5³ is 5 × 5 × ... with 3 copies, not 5 × 3.

Method. Multiply powers with the same base by adding the indices.

1. Check the bases are the same. The index laws only work when the base is the same.
   Both bases are a.
2. Add the indices. When powers with the same base are multiplied, add the indices.
   7 + 2 = 9
3. Write the answer.
   a⁹

Check. a appears 9 times altogether, so the index is 9.

Watch out. Add the indices. Do not multiply them, and do not multiply the bases.

Method. Divide powers with the same base by subtracting the indices.

1. Check the bases are the same. The index laws need the same base.
   Both bases are b.
2. Subtract the indices. When powers with the same base are divided, subtract the bottom index from the top index.
   10 − 4 = 6
3. Write the answer.
   b⁶

Check. b⁶ × b⁴ gives b¹⁰, so the answer is right.

Watch out. Subtract the bottom index from the top index. Keep the order: top minus bottom.

Method. Any number that is not zero, raised to the power 0, is 1.

1. Divide a power by itself. Dividing a number by itself always gives 1.
   9³ ÷ 9³ = 1
2. Use the division index law. Subtract the indices: 3 − 3 = 0.
   9³ ÷ 9³ = 9⁰
3. Write the answer. So the power to the zero is 1.
   9⁰ = 1

Check. Both routes give 1, so 9⁰ = 1 for any base that is not zero.

Watch out. Zero as an index means 1, not 0.

Method. A negative index means one over the positive power.

1. Look at the negative index. The minus sign tells us to take the reciprocal (flip the fraction).
   4⁻²
2. Use the rule. Write 1 over 4 to the positive power 2.
   1 ÷ 4²
3. Work out the power and write the answer. 4² is 16.
   1/16

Check. 1/16 is a small positive number, so a negative index does not give a negative answer.

Watch out. A negative index makes a small number. It does not make the answer negative.

Method. A fraction with 1 on top is a negative index. Flip the power to the top: 1 over 2⁶ is 2⁻⁶.

1. Find the power in the bottom of the fraction. The bottom is 2 to the power 6.
   1/64 = 1/2⁶
2. Move the power to the top. Flipping the power to the top gives a negative index.
   1/2⁶ = 2⁻⁶
3. Write the answer.
   2⁻⁶

Check. 2⁻⁶ = 1/64, which is the fraction we started with.

Watch out. The base in the bottom of the fraction is the base of the index.

Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places you moved.

1. Find the decimal point. For a whole number it is at the end. For a decimal it is after the units digit.
   0.000036
2. Move the decimal point 5 places right. The decimal point moves right for a small number, so the power is negative.
   3.6 with 5 moves
3. Write the answer in standard form. The power of 10 is -5.
   3.6 × 10⁻⁵

Check. 3.6 × 10⁻⁵ is 0.000036, which is the number we started with.

Watch out. The power counts the moves. Moving left gives a positive power. Moving right gives a negative power.

Method. Use the power of 10 to move the decimal point, then write the number in ordinary form.

1. Look at the power of 10. The power is 6.
   10⁶
2. Move the decimal point 6 places right. A positive power makes the number bigger, so the decimal point moves right.
   4.7 × 10⁶
3. Write the answer. Fill any gaps with zeros.
   4,700,000

Check. The answer should be bigger than 4.7, and 4,700,000 is.

Watch out. Fill any gaps with zeros. Count the places carefully.

Method. Multiply the front numbers, and add the powers of 10. Then check the front number is between 1 and 10.

1. Multiply the numbers in front. Multiply the two front numbers on their own.
   3 × 2 = 6
2. Add the powers of 10. When the base is 10 in both, add the indices.
   5 + 2 = 7
3. Write the answer in standard form.
   6 × 10⁷

Check. Ordinary numbers: 300,000 × 200 = 60,000,000.

Watch out. Do not add the front numbers. Only the powers of 10 are added.

Method. Divide the front numbers, and subtract the powers of 10. Then check the front number is between 1 and 10.

1. Divide the front numbers. Divide the first front number by the second.
   6 ÷ 2 = 3
2. Subtract the powers of 10. Subtract the bottom power from the top power.
   6 − 3 = 3
3. Write the answer in standard form.
   3 × 10³

Check. Ordinary numbers: 6,000,000 ÷ 2,000 = 3,000.

Watch out. Subtract the bottom power from the top power. Keep the order.

Method. Move the decimal point so that there is one non-zero digit in front of it. Then count the places you moved.

1. Find the decimal point. For a whole number it is at the end. For a decimal it is after the units digit.
   0.0000072
2. Move the decimal point 6 places right. The decimal point moves right for a small number, so the power is negative.
   7.2 with 6 moves
3. Write the answer in standard form. The power of 10 is -6.
   7.2 × 10⁻⁶

Check. 7.2 × 10⁻⁶ is 0.0000072, which is the number we started with.

Watch out. The power counts the moves. Moving left gives a positive power. Moving right gives a negative power.

Method. Add the indices: 5 + (−2) = 3.

1. Add the indices. 5 + (−2) = 3.
   p^{5 + (−2)} = p³
2. Write the answer.
   p³

Check. p⁵ ÷ p² = p³ gives the same result.

Method. Multiply the front numbers, and add the powers of 10. Then check the front number is between 1 and 10.

1. Multiply the numbers in front. Multiply the two front numbers on their own.
   4 × 3 = 12
2. Add the powers of 10. When the base is 10 in both, add the indices.
   9 + 0 = 9
3. Make the front number between 1 and 10. It must be 1 or more and less than 10. Move the point and change the power.
   12 × 10⁹ = 1.2 × 10¹⁰
4. Write the answer in standard form.
   1.2 × 10¹⁰

Check. Ordinary numbers: 4,000,000,000 × 3 = 12,000,000,000.

Watch out. If the front number is 10 or more, divide it by 10 and add 1 to the power.

Method. Divide the front numbers and subtract the powers of 10, then write the answer as an ordinary number.

1. Divide the front numbers. 1.6 ÷ 2 = 0.8.
   1.6 ÷ 2 = 0.8
2. Subtract the powers. 7 − 4 = 3.
   10^{7 − 4} = 10³
3. Combine. 0.8 × 10³ = 800.
   0.8 × 10³ = 800

Check. 800 × 2 × 10⁴ = 1.6 × 10⁷, which is right.

Method. Write all four numbers as ordinary decimals, then order them.

1. Write each number as a decimal. 3 × 10⁻² = 0.03
   0.03
2. Write each number as a decimal. 2.5 × 10⁻³ = 0.0025
   0.0025
3. Order the decimals. The smallest has the most zeros after the point.
   0.0003 < 0.0025 < 0.03 < 0.04
4. Write the answer in the original form. Put the original numbers in that order.
   0.0003, 2.5 × 10⁻³, 3 × 10⁻², 0.04

Check. Each number is between 0 and 1, so the one with the most zeros after the point is the smallest.

Method. Write both numbers in ordinary form, so they are easy to compare.

1. Write both numbers in ordinary form. Then the sizes are easy to see.
   3.6 × 10⁴ = 36,000 and 4.2 × 10³ = 4,200
2. Compare the sizes. The bigger number is the one with the bigger value.
   36,000 is the bigger number
3. Write the answer.
   3.6 × 10⁴

Check. A bigger power of 10 usually means a bigger number, but compare the numbers in front too when the powers match.

Watch out. Compare the whole numbers, not only the powers. A large front number can make a number bigger.

Method. Square the number and the power: (a × 10ⁿ)² = a² × 10²ⁿ.

1. Square the front number. 5² = 25.
   5² = 25
2. Multiply the power by 2. The power 3 is multiplied by 2, to give 6.
   10^{3 × 2} = 10⁶
3. Write in standard form. 25 is 2.5 times 10, so the power goes up by 1.
   25 × 10⁶ = 2.5 × 10⁷
4. Write the answer.
   2.5 × 10⁷

Check. 5000² = 25,000,000 = 2.5 × 10⁷.

Method. Divide a power by itself and use the index laws.

1. Write a division of equal powers. Any power divided by itself is 1.
   5³ ÷ 5³ = 1
2. Use the division index law. Subtract the indices: 3 − 3 = 0.
   5^{3 − 3} = 5⁰
3. Compare the two results. Both results are the same number, so 5⁰ must equal 1.
   5⁰ = 1
4. Write the answer.
   5⁰ = 1

Check. The same reasoning works for any base that is not zero.

Method. Multiply the numbers, then multiply the powers of a by adding the indices.

1. Multiply the numbers. 2 × 3 = 6.
   2 × 3 = 6
2. Add the indices of a. a³ × a⁴ = a^{3 + 4}.
   a³ × a⁴ = a⁷
3. Write the answer.
   6a⁷

Check. The number part is 6, and the index of a is 7.

Method. Multiply the front numbers, and add the powers of 10. Then check the front number is between 1 and 10.

1. Multiply the numbers in front. Multiply the two front numbers on their own.
   2 × 5 = 10
2. Add the powers of 10. When the base is 10 in both, add the indices.
   -6 + 3 = -3
3. Make the front number between 1 and 10. It must be 1 or more and less than 10. Move the point and change the power.
   10 × 10^-3 = 1 × 10^-2
4. Write the answer in standard form.
   1 × 10⁻²

Check. Ordinary numbers: 0.000002 × 5,000 = 0.01.

Watch out. If the front number is 10 or more, divide it by 10 and add 1 to the power.