GCSE Maths: Zero and negative indices (Higher)

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Any number (except 0) raised to the power 0 equals 1.

1. Write down the pattern of powers of 5, going down. Each time the index goes down by 1 we divide by 5.
   5³ = 125
   5² = 25
   5 = 5
2. Continue the pattern for index 0. Dividing 5 by 5 gives 1. This also follows from the division law: 5³ ÷ 5³ = 5⁰, and anything divided by itself is 1.
   5⁰ = 5 ÷ 5 = 1
3. Write the answer. The same works for every base, so the answer does not depend on the base.
   5⁰ = 1

Check. 5⁴ ÷ 5⁴ = 5⁰ and also = 1. ✓

Watch out. 5⁰ is 1, not 0 and not 5.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Any number (except 0) raised to the power 0 equals 1.

1. Write down the pattern of powers of 8, going down. Each time the index goes down by 1 we divide by 8.
   8³ = 512
   8² = 64
   8 = 8
2. Continue the pattern for index 0. Dividing 8 by 8 gives 1. This also follows from the division law: 8³ ÷ 8³ = 8⁰, and anything divided by itself is 1.
   8⁰ = 8 ÷ 8 = 1
3. Write the answer. The same works for every base, so the answer does not depend on the base.
   8⁰ = 1

Check. 8⁴ ÷ 8⁴ = 8⁰ and also = 1. ✓

Watch out. 8⁰ is 1, not 0 and not 8.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Any number (except 0) raised to the power 0 equals 1.

1. Write down the pattern of powers of 10, going down. Each time the index goes down by 1 we divide by 10.
   10³ = 1000
   10² = 100
   10 = 10
2. Continue the pattern for index 0. Dividing 10 by 10 gives 1. This also follows from the division law: 10³ ÷ 10³ = 10⁰, and anything divided by itself is 1.
   10⁰ = 10 ÷ 10 = 1
3. Write the answer. The same works for every base, so the answer does not depend on the base.
   10⁰ = 1

Check. 10⁴ ÷ 10⁴ = 10⁰ and also = 1. ✓

Watch out. 10⁰ is 1, not 0 and not 10.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Any number (except 0) raised to the power 0 equals 1.

1. Write down the pattern of powers of 12, going down. Each time the index goes down by 1 we divide by 12.
   12³ = 1728
   12² = 144
   12 = 12
2. Continue the pattern for index 0. Dividing 12 by 12 gives 1. This also follows from the division law: 12³ ÷ 12³ = 12⁰, and anything divided by itself is 1.
   12⁰ = 12 ÷ 12 = 1
3. Write the answer. The same works for every base, so the answer does not depend on the base.
   12⁰ = 1

Check. 12⁴ ÷ 12⁴ = 12⁰ and also = 1. ✓

Watch out. 12⁰ is 1, not 0 and not 12.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Any number (except 0) raised to the power 0 equals 1.

1. Write down the pattern of powers of 100, going down. Each time the index goes down by 1 we divide by 100.
   100³ = 1,000,000
   100² = 10,000
   100 = 100
2. Continue the pattern for index 0. Dividing 100 by 100 gives 1. This also follows from the division law: 100³ ÷ 100³ = 100⁰, and anything divided by itself is 1.
   100⁰ = 100 ÷ 100 = 1
3. Write the answer. The same works for every base, so the answer does not depend on the base.
   100⁰ = 1

Check. 100⁴ ÷ 100⁴ = 100⁰ and also = 1. ✓

Watch out. 100⁰ is 1, not 0 and not 100.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Any number (except 0) raised to the power 0 equals 1.

1. Write down the pattern of powers of 3, going down. Each time the index goes down by 1 we divide by 3.
   3³ = 27
   3² = 9
   3 = 3
2. Continue the pattern for index 0. Dividing 3 by 3 gives 1. This also follows from the division law: 3³ ÷ 3³ = 3⁰, and anything divided by itself is 1.
   3⁰ = 3 ÷ 3 = 1
3. Write the answer. The same works for every base, so the answer does not depend on the base.
   3⁰ = 1

Check. 3⁴ ÷ 3⁴ = 3⁰ and also = 1. ✓

Watch out. 3⁰ is 1, not 0 and not 3.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 2.
   2³ = 8
   2² = 4
   2 = 2
   2⁰ = 1
   2⁻¹ = 12
   2⁻² = 14
2. Spot the rule. A negative index means “one over”: 2⁻² is the reciprocal of 2².
   2⁻² = 12²
3. Work out the value of the bottom. Evaluate the positive power.
   2² = 4, so 2⁻² = 14

Check. 2⁻² × 2² should be 2⁰ = 1: 14 × 4 = 1. ✓

Watch out. 2⁻² is 14, not −4. The minus sign in the index changes it to a reciprocal.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 3.
   3³ = 27
   3² = 9
   3 = 3
   3⁰ = 1
   3⁻¹ = 13
   3⁻² = 19
2. Spot the rule. A negative index means “one over”: 3⁻² is the reciprocal of 3².
   3⁻² = 13²
3. Work out the value of the bottom. Evaluate the positive power.
   3² = 9, so 3⁻² = 19

Check. 3⁻² × 3² should be 3⁰ = 1: 19 × 9 = 1. ✓

Watch out. 3⁻² is 19, not −9. The minus sign in the index changes it to a reciprocal.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 5.
   5³ = 125
   5² = 25
   5 = 5
   5⁰ = 1
   5⁻¹ = 15
2. Spot the rule. A negative index means “one over”: 5⁻¹ is the reciprocal of 5.
   5⁻¹ = 15
3. Work out the value of the bottom. Evaluate the positive power.
   5 = 5, so 5⁻¹ = 15

Check. 5⁻¹ × 5 should be 5⁰ = 1: 15 × 5 = 1. ✓

Watch out. 5⁻¹ is 15, not −5. The minus sign in the index changes it to a reciprocal.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 10.
   10³ = 1000
   10² = 100
   10 = 10
   10⁰ = 1
   10⁻¹ = 110
   10⁻² = 1100
   10⁻³ = 11000
2. Spot the rule. A negative index means “one over”: 10⁻³ is the reciprocal of 10³.
   10⁻³ = 110³
3. Work out the value of the bottom. Evaluate the positive power.
   10³ = 1000, so 10⁻³ = 11000

Check. 10⁻³ × 10³ should be 10⁰ = 1: 11000 × 1000 = 1. ✓

Watch out. 10⁻³ is 11000, not −1000. The minus sign in the index changes it to a reciprocal.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 4.
   4³ = 64
   4² = 16
   4 = 4
   4⁰ = 1
   4⁻¹ = 14
   4⁻² = 116
2. Spot the rule. A negative index means “one over”: 4⁻² is the reciprocal of 4².
   4⁻² = 14²
3. Work out the value of the bottom. Evaluate the positive power.
   4² = 16, so 4⁻² = 116

Check. 4⁻² × 4² should be 4⁰ = 1: 116 × 16 = 1. ✓

Watch out. 4⁻² is 116, not −16. The minus sign in the index changes it to a reciprocal.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 2.
   2³ = 8
   2² = 4
   2 = 2
   2⁰ = 1
   2⁻¹ = 12
   2⁻² = 14
   2⁻³ = 18
   2⁻⁴ = 116
2. Spot the rule. A negative index means “one over”: 2⁻⁴ is the reciprocal of 2⁴.
   2⁻⁴ = 12⁴
3. Work out the value of the bottom. Evaluate the positive power.
   2⁴ = 16, so 2⁻⁴ = 116

Check. 2⁻⁴ × 2⁴ should be 2⁰ = 1: 116 × 16 = 1. ✓

Watch out. 2⁻⁴ is 116, not −16. The minus sign in the index changes it to a reciprocal.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 2.
   2³ = 8
   2² = 4
   2 = 2
   2⁰ = 1
   2⁻¹ = 12
   2⁻² = 14
   2⁻³ = 18
2. Spot the rule. A negative index means “one over”: 2⁻³ is the reciprocal of 2³.
   2⁻³ = 12³
3. Work out the value of the bottom. Evaluate the positive power.
   2³ = 8, so 2⁻³ = 18
4. Write it as a decimal as well. Divide 1 by 8.
   1 ÷ 8 = 0.125

Check. 2⁻³ × 2³ should be 2⁰ = 1: 18 × 8 = 1. ✓

Watch out. 2⁻³ is 18, not −8. The minus sign in the index changes it to a reciprocal.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 5.
   5³ = 125
   5² = 25
   5 = 5
   5⁰ = 1
   5⁻¹ = 15
   5⁻² = 125
2. Spot the rule. A negative index means “one over”: 5⁻² is the reciprocal of 5².
   5⁻² = 15²
3. Work out the value of the bottom. Evaluate the positive power.
   5² = 25, so 5⁻² = 125
4. Write it as a decimal as well. Divide 1 by 25.
   1 ÷ 25 = 0.04

Check. 5⁻² × 5² should be 5⁰ = 1: 125 × 25 = 1. ✓

Watch out. 5⁻² is 125, not −25. The minus sign in the index changes it to a reciprocal.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 10.
   10³ = 1000
   10² = 100
   10 = 10
   10⁰ = 1
   10⁻¹ = 110
   10⁻² = 1100
2. Spot the rule. A negative index means “one over”: 10⁻² is the reciprocal of 10².
   10⁻² = 110²
3. Work out the value of the bottom. Evaluate the positive power.
   10² = 100, so 10⁻² = 1100
4. Write it as a decimal as well. Divide 1 by 100.
   1 ÷ 100 = 0.01

Check. 10⁻² × 10² should be 10⁰ = 1: 1100 × 100 = 1. ✓

Watch out. 10⁻² is 1100, not −100. The minus sign in the index changes it to a reciprocal.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 4.
   4³ = 64
   4² = 16
   4 = 4
   4⁰ = 1
   4⁻¹ = 14
2. Spot the rule. A negative index means “one over”: 4⁻¹ is the reciprocal of 4.
   4⁻¹ = 14
3. Work out the value of the bottom. Evaluate the positive power.
   4 = 4, so 4⁻¹ = 14
4. Write it as a decimal as well. Divide 1 by 4.
   1 ÷ 4 = 0.25

Check. 4⁻¹ × 4 should be 4⁰ = 1: 14 × 4 = 1. ✓

Watch out. 4⁻¹ is 14, not −4. The minus sign in the index changes it to a reciprocal.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 10.
   10³ = 1000
   10² = 100
   10 = 10
   10⁰ = 1
   10⁻¹ = 110
2. Spot the rule. A negative index means “one over”: 10⁻¹ is the reciprocal of 10.
   10⁻¹ = 110
3. Work out the value of the bottom. Evaluate the positive power.
   10 = 10, so 10⁻¹ = 110
4. Write it as a decimal as well. Divide 1 by 10.
   1 ÷ 10 = 0.1

Check. 10⁻¹ × 10 should be 10⁰ = 1: 110 × 10 = 1. ✓

Watch out. 10⁻¹ is 110, not −10. The minus sign in the index changes it to a reciprocal.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together. The reciprocal of a number is 1 divided by it.

Method. A negative index means the reciprocal: a⁻ⁿ = 1 ÷ aⁿ. It does not make the answer negative.

1. Write down the pattern of powers going down. Each time the index goes down by 1 we divide by 2.
   2³ = 8
   2² = 4
   2 = 2
   2⁰ = 1
   2⁻¹ = 12
   2⁻² = 14
2. Spot the rule. A negative index means “one over”: 2⁻² is the reciprocal of 2².
   2⁻² = 12²
3. Work out the value of the bottom. Evaluate the positive power.
   2² = 4, so 2⁻² = 14
4. Write it as a decimal as well. Divide 1 by 4.
   1 ÷ 4 = 0.25

Check. 2⁻² × 2² should be 2⁰ = 1: 14 × 4 = 1. ✓

Watch out. 2⁻² is 14, not −4. The minus sign in the index changes it to a reciprocal.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the number under the 1 as a power of 3, then use a negative index.

1. Write 9 as a power of 3. Multiply 3 by itself until you reach 9.
   9 = 3²
2. Use the rule for negative indices. 1 divided by a power is that power with a negative index.
   19 = 13²
3. Write the answer.
   19 = 3⁻²

Check. 3⁻² = 13² = 19. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the number under the 1 as a power of 2, then use a negative index.

1. Write 8 as a power of 2. Multiply 2 by itself until you reach 8.
   8 = 2³
2. Use the rule for negative indices. 1 divided by a power is that power with a negative index.
   18 = 12³
3. Write the answer.
   18 = 2⁻³

Check. 2⁻³ = 12³ = 18. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the number under the 1 as a power of 5, then use a negative index.

1. Write 25 as a power of 5. Multiply 5 by itself until you reach 25.
   25 = 5²
2. Use the rule for negative indices. 1 divided by a power is that power with a negative index.
   125 = 15²
3. Write the answer.
   125 = 5⁻²

Check. 5⁻² = 15² = 125. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the number under the 1 as a power of 10, then use a negative index.

1. Write 100 as a power of 10. Multiply 10 by itself until you reach 100.
   100 = 10²
2. Use the rule for negative indices. 1 divided by a power is that power with a negative index.
   1100 = 110²
3. Write the answer.
   1100 = 10⁻²

Check. 10⁻² = 110² = 1100. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the number under the 1 as a power of 2, then use a negative index.

1. Write 32 as a power of 2. Multiply 2 by itself until you reach 32.
   32 = 2⁵
2. Use the rule for negative indices. 1 divided by a power is that power with a negative index.
   132 = 12⁵
3. Write the answer.
   132 = 2⁻⁵

Check. 2⁻⁵ = 12⁵ = 132. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the number under the 1 as a power of 3, then use a negative index.

1. Write 81 as a power of 3. Multiply 3 by itself until you reach 81.
   81 = 3⁴
2. Use the rule for negative indices. 1 divided by a power is that power with a negative index.
   181 = 13⁴
3. Write the answer.
   181 = 3⁻⁴

Check. 3⁻⁴ = 13⁴ = 181. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the same index laws as before. They work for negative indices too.

1. Check the bases are the same. Both powers have base 2, so we can use the index law.
   2⁵ × 2⁻²
2. Add the indices. Multiplying: add the indices. Dividing: subtract the indices. Take care with the negative signs.
   5 + (−2) = 3
   2⁵ × 2⁻² = 2³
3. Work out the value. Multiply the copies out.
   2³ = 8

Check. Work out each power as a number first, then multiply or divide: 32 × 14 = 8. ✓

Watch out. Be careful when you subtract a negative index: 2³ ÷ 2⁻² = 2^{3 − (−2)} = 2⁵.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the same index laws as before. They work for negative indices too.

1. Check the bases are the same. Both powers have base 3, so we can use the index law.
   3⁻¹ × 3⁴
2. Add the indices. Multiplying: add the indices. Dividing: subtract the indices. Take care with the negative signs.
   −1 + 4 = 3
   3⁻¹ × 3⁴ = 3³
3. Work out the value. Multiply the copies out.
   3³ = 27

Check. Work out each power as a number first, then multiply or divide: 13 × 81 = 27. ✓

Watch out. Be careful when you subtract a negative index: 2³ ÷ 2⁻² = 2^{3 − (−2)} = 2⁵.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the same index laws as before. They work for negative indices too.

1. Check the bases are the same. Both powers have base 5, so we can use the index law.
   5² ÷ 5⁴
2. Subtract the indices. Multiplying: add the indices. Dividing: subtract the indices. Take care with the negative signs.
   2 − 4 = −2
   5² ÷ 5⁴ = 5⁻²
3. Write the negative index as a fraction. A negative index means 1 divided by the positive power.
   5⁻² = 15² = 125

Check. Work out each power as a number first, then multiply or divide: 25 ÷ 625 = 125. ✓

Watch out. Be careful when you subtract a negative index: 2³ ÷ 2⁻² = 2^{3 − (−2)} = 2⁵.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the same index laws as before. They work for negative indices too.

1. Check the bases are the same. Both powers have base 10, so we can use the index law.
   10³ × 10⁻⁵
2. Add the indices. Multiplying: add the indices. Dividing: subtract the indices. Take care with the negative signs.
   3 + (−5) = −2
   10³ × 10⁻⁵ = 10⁻²
3. Write the negative index as a fraction. A negative index means 1 divided by the positive power.
   10⁻² = 110² = 1100

Check. Work out each power as a number first, then multiply or divide: 1000 × 1100000 = 1100. ✓

Watch out. Be careful when you subtract a negative index: 2³ ÷ 2⁻² = 2^{3 − (−2)} = 2⁵.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the same index laws as before. They work for negative indices too.

1. Check the bases are the same. Both powers have base 2, so we can use the index law.
   2⁻³ ÷ 2⁻⁵
2. Subtract the indices. Multiplying: add the indices. Dividing: subtract the indices. Take care with the negative signs.
   −3 − (−5) = 2
   2⁻³ ÷ 2⁻⁵ = 2²
3. Work out the value. Multiply the copies out.
   2² = 4

Check. Work out each power as a number first, then multiply or divide: 18 ÷ 132 = 4. ✓

Watch out. Be careful when you subtract a negative index: 2³ ÷ 2⁻² = 2^{3 − (−2)} = 2⁵.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the same index laws as before. They work for negative indices too.

1. Check the bases are the same. Both powers have base 4, so we can use the index law.
   4⁻² × 4²
2. Add the indices. Multiplying: add the indices. Dividing: subtract the indices. Take care with the negative signs.
   −2 + 2 = 0
   4⁻² × 4² = 4⁰
3. Use the zero index. Anything to the power 0 is 1.
   4⁰ = 1

Check. Work out each power as a number first, then multiply or divide: 116 × 16 = 1. ✓

Watch out. Be careful when you subtract a negative index: 2³ ÷ 2⁻² = 2^{3 − (−2)} = 2⁵.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the rule for negative indices.

1. Write the negative index as a reciprocal. a⁻ⁿ = 1 ÷ aⁿ.
   3⁻² = 1 ÷ 3²
2. Work out 3².
   3² = 9, so 3⁻² = 19
3. Compare. A negative index does not make the answer negative.
   −9 is wrong, so the statement is false.

Check. 3² × 3⁻² = 9 × 19 = 1 = 3⁰. ✓

Watch out. The minus sign in the index means reciprocal.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write it as a fraction and then a decimal.

1. Use the negative index rule. a⁻ⁿ = 1 ÷ aⁿ.
   2⁻¹ = 12
2. Change to a decimal. A half is 0.5.
   12 = 0.5

Check. 0.5 × 2 = 1. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the zero index rule.

1. Recall the rule. Any number (except 0) to the power 0 is 1.
   10⁰ = 10³ ÷ 10³ = 1000 ÷ 1000
2. Work it out.
   = 1

Check. The pattern 1000, 100, 10, 1 gives 10⁰ = 1. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Use the negative index rule.

1. Write as a reciprocal.
   4⁻² = 1 ÷ 4²
2. Work out 4².
   4² = 16, so the answer is 116

Check. 4² × 116 = 1. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. For a fraction with a negative index: flip the fraction, then use the positive index.

1. Use the negative index: take the reciprocal of the fraction. A negative index flips the fraction upside down and the index becomes positive.
   (23)⁻² = (32)²
2. Raise the top and the bottom to the power. The power applies to the numerator and to the denominator: 3² on the top and 2² on the bottom.
   3²2² = 94
3. Write the answer. 2 and 3 share no common factor, so the fraction is already in its simplest form. It is bigger than 1 because the original fraction was less than 1.
   (23)⁻² = 94

Check. (23)⁻² × (23)² should be 1: 94 × 49 = 1. ✓

Watch out. Flip the whole fraction (both top and bottom). Do not make the answer negative.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. For a fraction with a negative index: flip the fraction, then use the positive index.

1. Use the negative index: take the reciprocal of the fraction. A negative index flips the fraction upside down and the index becomes positive.
   (25)⁻² = (52)²
2. Raise the top and the bottom to the power. The power applies to the numerator and to the denominator: 5² on the top and 2² on the bottom.
   5²2² = 254
3. Write the answer. 2 and 5 share no common factor, so the fraction is already in its simplest form. It is bigger than 1 because the original fraction was less than 1.
   (25)⁻² = 254

Check. (25)⁻² × (25)² should be 1: 254 × 425 = 1. ✓

Watch out. Flip the whole fraction (both top and bottom). Do not make the answer negative.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. For a fraction with a negative index: flip the fraction, then use the positive index.

1. Use the negative index: take the reciprocal of the fraction. A negative index flips the fraction upside down and the index becomes positive.
   (34)⁻³ = (43)³
2. Raise the top and the bottom to the power. The power applies to the numerator and to the denominator: 4³ on the top and 3³ on the bottom.
   4³3³ = 6427
3. Write the answer. 3 and 4 share no common factor, so the fraction is already in its simplest form. It is bigger than 1 because the original fraction was less than 1.
   (34)⁻³ = 6427

Check. (34)⁻³ × (34)³ should be 1: 6427 × 2764 = 1. ✓

Watch out. Flip the whole fraction (both top and bottom). Do not make the answer negative.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. For a fraction with a negative index: flip the fraction, then use the positive index.

1. Use the negative index: take the reciprocal of the fraction. A negative index flips the fraction upside down and the index becomes positive.
   (23)⁻³ = (32)³
2. Raise the top and the bottom to the power. The power applies to the numerator and to the denominator: 3³ on the top and 2³ on the bottom.
   3³2³ = 278
3. Write the answer. 2 and 3 share no common factor, so the fraction is already in its simplest form. It is bigger than 1 because the original fraction was less than 1.
   (23)⁻³ = 278

Check. (23)⁻³ × (23)³ should be 1: 278 × 827 = 1. ✓

Watch out. Flip the whole fraction (both top and bottom). Do not make the answer negative.

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Work out each power as a fraction, then add the fractions.

1. Write each power as a fraction. Negative index: reciprocal.
   2⁻¹ = 12 and 2⁻² = 14
2. Give both fractions the same denominator. The common denominator is 4.
   12 = 24
3. Add. Add the numerators and keep the denominator.
   24 + 14 = 34

Check. 0.5 + 0.25 = 0.75 = 34. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Work out each power, then add.

1. Work out each power. Zero index is 1; negative index is a reciprocal.
   3⁰ = 1 and 3⁻¹ = 13
2. Add the whole number and the fraction. Write 1 as 33 first.
   1 + 13 = 33 + 13 = 43 = 113

Check. 1 + 0.333... = 1.333... = 43. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the fraction as a power of the base, using a negative index.

1. Write what we want. The number 1/32 is smaller than 1, so the index will be negative.
   2ⁿ = 132
2. Write 32 as a power of 2. Multiply by the base until you reach it.
   32 = 2⁵
3. Use a negative index.
   132 = 2⁻⁵, so n = −5

Check. 2⁻⁵ = 12⁵ = 132. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the fraction as a power of the base, using a negative index.

1. Write what we want. The number 1/81 is smaller than 1, so the index will be negative.
   3ⁿ = 181
2. Write 81 as a power of 3. Multiply by the base until you reach it.
   81 = 3⁴
3. Use a negative index.
   181 = 3⁻⁴, so n = −4

Check. 3⁻⁴ = 13⁴ = 181. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the fraction as a power of the base, using a negative index.

1. Write what we want. The number 1/125 is smaller than 1, so the index will be negative.
   5ⁿ = 1125
2. Write 125 as a power of 5. Multiply by the base until you reach it.
   125 = 5³
3. Use a negative index.
   1125 = 5⁻³, so n = −3

Check. 5⁻³ = 15³ = 1125. ✓

Key words. In 2⁵, 2 is the base and 5 is the index (also called the power or exponent). The plural of index is indices. 2⁵ means five 2s multiplied together.

Method. Write the fraction as a power of the base, using a negative index.

1. Write what we want. The number 1/1000 is smaller than 1, so the index will be negative.
   10ⁿ = 11000
2. Write 1000 as a power of 10. Multiply by the base until you reach it.
   1000 = 10³
3. Use a negative index.
   11000 = 10⁻³, so n = −3

Check. 10⁻³ = 110³ = 11000. ✓