Year 3 · Unit 1 · Lesson 1 of 5

Zero and Negative Indices: What Is 2⁻³?

About 15 minutes · the zero index · negative indices as reciprocals · powers of 10 · fractions to negative powers · index laws with negative indices · no calculator · exit quiz

Hook

Look at the pattern 2³ = 8, 2² = 4, 2¹ = 2. Each time the index goes down by 1, the value halves.

Keep going. What are 2⁰, 2⁻¹ and 2⁻³?

Halving 2 gives 2⁰ = 1. Halving again gives 2⁻¹ = 12, then 2⁻² = 14, and 2⁻³ = 18. A negative index does not make a number negative. It means "one over": 2⁻³ = 1 ÷ 2³.

PowerValue
2³8
2²4÷ 2
2¹2÷ 2
2⁰1÷ 2
2⁻¹12÷ 2
2⁻²14÷ 2
2⁻³18÷ 2

Each step down the index, the value halves.

Learn

Master the essentials

IndexMeaning · example
Zeroa⁰ = 1
7⁰ = 1
Negativea−n = 1 ÷ an
6⁻² = 136
Fraction baseflip the fraction
(13)⁻² = 3² = 9
Powers of 10decimal places
10⁻² = 0.01

A negative index gives a reciprocal, never a negative number.

The zero index

Any number (except 0) to the power 0 is 1.

7⁰ = 1 · 250⁰ = 1

Negative indices

a−n means 1 ÷ an: the reciprocal of the positive power.

6⁻² = 1 ÷ 6² = 136

Still positive

A negative index gives a small positive number, not a negative one.

4⁻¹ = 14, not −4

Powers of 10

A negative power of 10 is a decimal with that many places.

10⁻² = 0.01 · 10⁻⁴ = 0.0001

Fractions

A negative index flips the fraction, then use the positive power.

(13)⁻² = 3² = 9

Index laws still work

Add the indices to multiply and subtract them to divide. Watch the signs.

6⁻⁴ × 6⁶ = 6²

Worked examples

See it done

1. Work out 2⁻³ as a fraction.

  1. A negative index means one over: 2⁻³ = 1 ÷ 2³.
  2. 2³ = 8, so the answer is 18.

18

2. Write 10⁻³ as a decimal.

  1. 10⁻³ = 1 ÷ 10³ = 1 ÷ 1,000.
  2. That is three decimal places: 0.001.

0.001

3. Simplify 7² × 7⁻⁴. Write the answer as a fraction.

  1. Same base, multiplying: add the indices. 2 + (−4) = −2.
  2. 7⁻² = 1 ÷ 7² = 149.

149

Common mistake: Thinking a negative index makes the number negative. 2⁻³ is not −8 and not −6. It is one over 2³: 18, a small positive number.

PractiseNo calculator

10 questions with instant feedback

Choose the correct answer. Remember: a negative index means "one over". Questions 1–3 are warm-ups, 4–7 are at exam standard and 8–10 are challenge questions. After you answer, the feedback shows the full solution.

Score: 0 / 10Answered: 0 / 10

Warm-up (The basic skill: one or two steps.)

1.6⁰ =

Not quite. 6⁰ is not 6 × 0.

Solution

  1. 6² ÷ 6² = 1, and by the index law it is also 6⁰.
  2. So any number (except 0) to the power 0 is 1.

Answer: C. 1.

Not quite. The index 0 does not leave the number unchanged.

Solution

  1. 6² ÷ 6² = 1, and by the index law it is also 6⁰.
  2. So any number (except 0) to the power 0 is 1.

Answer: C. 1.

Correct.

Solution

  1. 6² ÷ 6² = 1, and by the index law it is also 6⁰.
  2. So any number (except 0) to the power 0 is 1.

Not quite. The index is not a digit to write after the 6.

Solution

  1. 6² ÷ 6² = 1, and by the index law it is also 6⁰.
  2. So any number (except 0) to the power 0 is 1.

Answer: C. 1.

2.4⁻¹ =

Not quite. A negative index does not make the number negative.

Solution

  1. A negative index means one over: 4⁻¹ = 1 ÷ 4¹.
  2. 1 ÷ 4 = 14.

Answer: C. 14.

Not quite. The answer is positive: one over 4.

Solution

  1. A negative index means one over: 4⁻¹ = 1 ÷ 4¹.
  2. 1 ÷ 4 = 14.

Answer: C. 14.

Correct.

Solution

  1. A negative index means one over: 4⁻¹ = 1 ÷ 4¹.
  2. 1 ÷ 4 = 14.

Not quite. Do not subtract 1 from the base.

Solution

  1. A negative index means one over: 4⁻¹ = 1 ÷ 4¹.
  2. 1 ÷ 4 = 14.

Answer: C. 14.

3.5⁻² =

Not quite. A negative index gives a positive answer.

Solution

  1. 5⁻² = 1 ÷ 5².
  2. 5² = 25, so the answer is 125.

Answer: C. 125.

Not quite. Do not multiply 5 by −2.

Solution

  1. 5⁻² = 1 ÷ 5².
  2. 5² = 25, so the answer is 125.

Answer: C. 125.

Correct.

Solution

  1. 5⁻² = 1 ÷ 5².
  2. 5² = 25, so the answer is 125.

Not quite. 5² is 25, not 10.

Solution

  1. 5⁻² = 1 ÷ 5².
  2. 5² = 25, so the answer is 125.

Answer: C. 125.

Exam standard (Several steps, like real exam questions.)

4.Write 10⁻⁴ as a decimal.

Not quite. A negative index gives a small positive number.

Solution

  1. 10⁻⁴ = 1 ÷ 10⁴ = 1 ÷ 10,000.
  2. That has four decimal places: 0.0001.

Answer: D. 0.0001.

Not quite. 10⁻⁴ has four decimal places, not five.

Solution

  1. 10⁻⁴ = 1 ÷ 10⁴ = 1 ÷ 10,000.
  2. That has four decimal places: 0.0001.

Answer: D. 0.0001.

Not quite. Do not multiply 10 by −4.

Solution

  1. 10⁻⁴ = 1 ÷ 10⁴ = 1 ÷ 10,000.
  2. That has four decimal places: 0.0001.

Answer: D. 0.0001.

Correct.

Solution

  1. 10⁻⁴ = 1 ÷ 10⁴ = 1 ÷ 10,000.
  2. That has four decimal places: 0.0001.

5.Simplify 7⁻³ × 7⁵. Leave your answer as a power.

Correct.

Solution

  1. Same base, multiplying: add the indices.
  2. −3 + 5 = 2, so the answer is 7².

Not quite. Multiplying powers: add the indices, don't multiply them.

Solution

  1. Same base, multiplying: add the indices.
  2. −3 + 5 = 2, so the answer is 7².

Answer: A. 7².

Not quite. −3 + 5 = 2, not 3 + 5.

Solution

  1. Same base, multiplying: add the indices.
  2. −3 + 5 = 2, so the answer is 7².

Answer: A. 7².

Not quite. Watch the sign: −3 + 5 = +2.

Solution

  1. Same base, multiplying: add the indices.
  2. −3 + 5 = 2, so the answer is 7².

Answer: A. 7².

6.(12)⁻² =

Not quite. A negative index flips the fraction first: ½ becomes 2.

Solution

  1. A negative index flips the fraction: ½ becomes 2.
  2. Then use the positive power: 2² = 4.

Answer: B. 4.

Correct.

Solution

  1. A negative index flips the fraction: ½ becomes 2.
  2. Then use the positive power: 2² = 4.

Not quite. A negative index does not make the answer negative.

Solution

  1. A negative index flips the fraction: ½ becomes 2.
  2. Then use the positive power: 2² = 4.

Answer: B. 4.

Not quite. Do not multiply ½ by −2.

Solution

  1. A negative index flips the fraction: ½ becomes 2.
  2. Then use the positive power: 2² = 4.

Answer: B. 4.

7.Write 127 as a power of 3.

Correct.

Solution

  1. 27 = 3 × 3 × 3 = 3³.
  2. One over 3³ is 3⁻³, so 127 = 3⁻³.

Not quite. 3³ is 27. One over 27 needs a negative index.

Solution

  1. 27 = 3 × 3 × 3 = 3³.
  2. One over 3³ is 3⁻³, so 127 = 3⁻³.

Answer: A. 3⁻³.

Not quite. 27 = 3³, not 3⁹.

Solution

  1. 27 = 3 × 3 × 3 = 3³.
  2. One over 3³ is 3⁻³, so 127 = 3⁻³.

Answer: A. 3⁻³.

Not quite. (−3)³ is −27, not one over 27.

Solution

  1. 27 = 3 × 3 × 3 = 3³.
  2. One over 3³ is 3⁻³, so 127 = 3⁻³.

Answer: A. 3⁻³.

Challenge (Harder problems that mix ideas or need reasoning.)

8.4⁻² + 2⁻³ =

Not quite. Do not add the tops and the bottoms. Use a common denominator, 16.

Solution

  1. 4⁻² = 116 and 2⁻³ = 18.
  2. Use a common denominator: 18 = 216.
  3. 116 + 216 = 316.

Answer: B. 316.

Correct.

Solution

  1. 4⁻² = 116 and 2⁻³ = 18.
  2. Use a common denominator: 18 = 216.
  3. 116 + 216 = 316.

Not quite. Negative indices give fractions, not negative numbers.

Solution

  1. 4⁻² = 116 and 2⁻³ = 18.
  2. Use a common denominator: 18 = 216.
  3. 116 + 216 = 316.

Answer: B. 316.

Not quite. One eighth is two sixteenths: 1/16 + 2/16.

Solution

  1. 4⁻² = 116 and 2⁻³ = 18.
  2. Use a common denominator: 18 = 216.
  3. 116 + 216 = 316.

Answer: B. 316.

9.Simplify x⁴ × x⁻⁶ ÷ x⁻³.

Not quite. Dividing by x⁻³ means subtract −3, which is the same as adding 3.

Solution

  1. Multiply first: add the indices, 4 + (−6) = −2.
  2. Then divide: subtract the indices, −2 − (−3) = −2 + 3 = 1.
  3. x¹ is just x.

Answer: D. x.

Not quite. Watch the signs: 4 + (−6) = −2.

Solution

  1. Multiply first: add the indices, 4 + (−6) = −2.
  2. Then divide: subtract the indices, −2 − (−3) = −2 + 3 = 1.
  3. x¹ is just x.

Answer: D. x.

Not quite. −2 − (−3) = −2 + 3 = +1, not −1.

Solution

  1. Multiply first: add the indices, 4 + (−6) = −2.
  2. Then divide: subtract the indices, −2 − (−3) = −2 + 3 = 1.
  3. x¹ is just x.

Answer: D. x.

Correct.

Solution

  1. Multiply first: add the indices, 4 + (−6) = −2.
  2. Then divide: subtract the indices, −2 − (−3) = −2 + 3 = 1.
  3. x¹ is just x.

10.Which of these is the largest? 3⁻², 5⁻¹, 4⁻¹, 10⁻¹

Not quite. 10⁻¹ = 110: a bigger base gives a smaller fraction.

Solution

  1. Write each as a fraction: 19, 15, 14 and 110.
  2. All the tops are 1, so the smallest bottom gives the largest fraction.
  3. The largest is 14 = 4⁻¹.

Answer: B. 4⁻¹.

Correct.

Solution

  1. Write each as a fraction: 19, 15, 14 and 110.
  2. All the tops are 1, so the smallest bottom gives the largest fraction.
  3. The largest is 14 = 4⁻¹.

Not quite. 3⁻² = 19, which is smaller than 14.

Solution

  1. Write each as a fraction: 19, 15, 14 and 110.
  2. All the tops are 1, so the smallest bottom gives the largest fraction.
  3. The largest is 14 = 4⁻¹.

Answer: B. 4⁻¹.

Not quite. 5⁻¹ = 15, which is smaller than 14.

Solution

  1. Write each as a fraction: 19, 15, 14 and 110.
  2. All the tops are 1, so the smallest bottom gives the largest fraction.
  3. The largest is 14 = 4⁻¹.

Answer: B. 4⁻¹.

Apply

Tiny measurements

Scientists use powers of 10 for tiny lengths. 1 millimetre (mm) = 10⁻³ m and 1 micrometre (µm) = 10⁻⁶ m.

1. Write 1 µm as a decimal of a metre.

2. How many micrometres make 1 millimetre? Give your answer as a power of 10 and as a number.

3. A bacterium is 4 µm long. How many fit end to end across 1 mm?

  1. 1 ÷ 1,000,000: 0.000001 m.
  2. 10⁻³ ÷ 10⁻⁶: subtract the indices, −3 − (−6) = 3, so 10³ = 1,000.
  3. 1,000 ÷ 4 = 250 bacteria.

Next: the Lesson 1 exit quiz. Five questions, recorded on your dashboard. Aim for 4 out of 5.

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