Negative exponents and the zero exponent

A negative exponent means divide instead of multiply: 2⁻³ = 1/8. Any number except 0 to the power of 0 is 1.

The idea: keep the pattern going

Write the powers of 2 going down: 2³ = 8, 2² = 4, 2¹ = 2. Each time the exponent drops by 1, the answer is divided by 2.

Keep going and the pattern decides the answers. 2⁰ = 2 ÷ 2 = 1, then 2⁻¹ = 1/2, and 2⁻² = 1/4. Nobody made these up: they are the only answers that keep the pattern.

See it: pictures to draw with your child

Copy these onto paper or a whiteboard. A quick sketch makes the idea concrete before any written method.

Each step down: ÷ 22³=8÷ 22²=4÷ 22¹=2÷ 22⁰=1÷ 22⁻¹=1/2÷ 22⁻²=1/4÷ 22⁻³=1/8
Each step down divides by 2, so 2⁰ = 1 and 2⁻³ = 1/8.
Each step down: ÷ 1010²=100÷ 1010¹=10÷ 1010⁰=1÷ 1010⁻¹=1/10÷ 1010⁻²=1/100÷ 1010⁻³=1/1,000
Negative powers of 10 are tenths, hundredths and thousandths.
5³ ÷ 5³= 125 ÷ 125 = 1rule: 5³⁻³ = 5⁰so 5⁰ = 1
The quotient rule and the pattern both give 1.

The zero exponent: always 1

Any number except 0 to the power of 0 is 1. So 5⁰ = 1, 100⁰ = 1 and (−7)⁰ = 1.

The quotient rule agrees. 5³ ÷ 5³ = 1, and the rule says 5³⁻³ = 5⁰. 0⁰ is left undefined in school maths.

Negative exponents: 1 over the power

A negative exponent means 1 divided by the same power with a positive exponent. 3⁻² = 1/3² = 1/9.

The answer is a small positive number, not a negative number. The minus sign in the exponent means “divide”, not “below zero”.

See the pattern table

The table under the answer shows each step dividing by the base. Try base 10 with exponent −4.

Negative powers of 10 are decimals

10⁻¹ = 1/10 = 0.1, 10⁻² = 1/100 = 0.01 and 10⁻³ = 0.001. The exponent tells you which decimal place the 1 is in.

This is how scientists write tiny numbers. A grain of sand about 0.0005 m wide is 5 × 10⁻⁴ m. See scientific notation.

Negative exponents with fractions

A negative exponent flips a fraction. (1/2)⁻¹ = 2, and (2/3)⁻² = (3/2)² = 9/4.

The rule is the same: 1 divided by the power. Dividing by a fraction is the same as multiplying by its flip, so the fraction turns over.

Key words

Teaching tip: build the table together

Start with a power your child knows, like 2⁴ = 16. Write the powers going down, dividing by 2 each time, all the way to 2⁻³.

Ask them to say the rule they see. Then ask what 3⁰ and 3⁻¹ must be, using a new table. Seeing the pattern twice is usually enough.

If your child later writes 2⁻³ = −8, ask them to rebuild the table. The mistake fixes itself.

A week of practice

Ten minutes a day is plenty. Short, regular practice beats one long session at the weekend.

If one day goes badly, repeat it rather than moving on. Each day builds on the one before.

By the end of the week your child should be able to explain, not just state, why 7⁰ = 1. Explaining it is the best sign the idea has stuck.

Where negative exponents are used

Scientists measure tiny things with negative powers of 10. A millimetre is 10⁻³ m and a micrometre is 10⁻⁶ m, about the size of a bacterium.

Computer storage uses powers of 2. Half a kilobyte is 2⁻¹ of a kilobyte, and the same pattern table explains why.

So the pattern on this page is not just a classroom rule. It is how very small amounts are written everywhere in science and technology.

Quiz: zero and negative exponents

Write it as a fraction or a decimal.

Where this fits: exponents step by step

  1. What are exponents
  2. Square numbers
  3. Square roots
  4. Estimating square roots
  5. Cube numbers and cube roots
  6. Powers of 10
  7. Exponent rules
  8. Zero and negative exponents (this page)
  9. Scientific notation
  10. Operations with scientific notation

What to say to your child

A short script for teaching it the first time. Pause after each line and let your child do the next step.

  1. “What is 2³?” (8.) “And 2²?” (4.)
  2. “What do we do each time we go down?” (Divide by 2.)
  3. “So what is 2⁰?” (1.)
  4. “And 2⁻¹?” (A half.)
  5. “Is 2⁻³ negative?” (No, it is 1/8, a small positive number.)

More worked examples

Example 1: zero exponent

9⁰ = 1.

Example 2: negative exponent

5⁻² = 1/5² = 1/25.

Example 3: power of 10

10⁻³ = 1/1,000 = 0.001.

Example 4: with the rules

2⁵ × 2⁻² = 2³ = 8.

Practice questions (answers below)

Ask these out loud or write them down. Check each answer, and ask “how did you get that?” for one or two.

  1. 2⁻¹
  2. 5⁻²
  3. 10⁻³
  4. 4⁰
  5. (−7)⁰
  6. 3⁻³
  7. 2⁻⁴
  8. Write 1/8 as a power of 2.
  9. Write 0.01 as a power of 10.
  10. Stretch: (1/2)⁻²
Show the answers
  1. 1/2
  2. 1/25
  3. 0.001
  4. 1
  5. 1
  6. 1/27
  7. 1/16
  8. 2⁻³
  9. 10⁻²
  10. 4

Common mistakes

Negative answerWrites 2⁻³ = −8.Fix: a negative exponent means 1 over the power: 1/8.
Zero exponent is zeroWrites 6⁰ = 0.Fix: any number except 0 to the power 0 is 1.
Multiplying by the exponentWrites 3⁻² = −6.Fix: 3⁻² = 1/3² = 1/9.

How to teach it at home

Halving chain

Start at 16 and keep halving: 8, 4, 2, 1, ½, ¼. Write each as a power of 2.

Decimal places

Say a power of 10 like 10⁻². Your child writes it as a decimal: 0.01.

Spot the trap

Write 5⁰ = 0 and 2⁻¹ = −2. Your child explains what is wrong.

By grade

More for each grade: Grade 8 math and the CBSE class 8 maths guide.

Worksheets and practice

Negative exponents worksheet, page 1

Negative exponents worksheet: printable PDF with an answer key. · Make a fresh test

Exponents practice · exponent calculator · exponents for kids · exponent rules worksheet · scientific notation worksheet · order of operations with exponents

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Questions parents ask

What does a negative exponent mean?

1 divided by the power with a positive exponent. 2⁻³ = 1/2³ = 1/8.

Why is any number to the power of 0 equal to 1?

Each step down in the exponent divides by the base. 2¹ = 2, so 2⁰ = 2 ÷ 2 = 1.

Is a negative exponent a negative number?

No. 2⁻³ is 1/8, a small positive number.

What is 10 to the power of −2?

1/100, which is 0.01.

What grade learns negative exponents?

Grade 8 under Common Core 8.EE.A.1.