somokai.

Powers and the laws of exponents

A power is repeated multiplication: 2⁵ = 2 · 2 · 2 · 2 · 2 = 32. With the same base, we add the exponents when we multiply and subtract them when we divide. The exponent zero gives 1, and 2⁻³ = 1/8.

Years 7–9 and upper secondary · Updated

Watch Fox work it out on the board

Fox works one step at a time. Press play.

Solve your own task

Write the task. You get one step at a time.

Base and exponent

In 2⁵, two is the base and five is the exponent. The exponent says how many twos are multiplied together.

2⁵ = 2 · 2 · 2 · 2 · 2 = 32

  • 3⁴ = 3 · 3 · 3 · 3 = 81
  • (−2)⁴ = 16: an even number of minus signs gives plus
  • (−2)³ = −8: an odd number of minus signs gives minus
  • −2⁴ = −(2 · 2 · 2 · 2) = −16: the exponent belongs to the two

The laws of exponents and why they hold

We write out the factors and see why the laws hold.

Three twos times four twos is seven twos:

2³ · 2⁴ = (2 · 2 · 2) · (2 · 2 · 2 · 2) = 2⁷ = 128

Two twos in the denominator cancel two twos in the numerator:

2⁶/2² = (2 · 2 · 2 · 2 · 2 · 2)/(2 · 2) = 2⁴ = 16

2³ twice is six twos:

(2³)² = 2³ · 2³ = 2⁶ = 64

LawExample
aᵐ · aⁿ = aᵐ⁺ⁿ5² · 5³ = 5⁵
aᵐ/aⁿ = aᵐ⁻ⁿ7⁶/7⁴ = 7²
(aᵐ)ⁿ = aᵐⁿ(3²)⁴ = 3⁸
(a · b)ⁿ = aⁿ · bⁿ(2 · 5)³ = 2³ · 5³ = 8 · 125 = 1 000

The laws apply to multiplication and division. 2³ + 2⁴ = 8 + 16 = 24, not 2⁷.

Negative exponents and standard form

Every time the exponent goes down by one, we divide by two.

PowerValue
2³8
2²4
2¹2
2⁰1
2⁻¹1/2
2⁻²1/4
2⁻³1/8

a⁰ = 1 a⁻ⁿ = 1/aⁿ

In standard form, we write a number that is at least 1 but less than 10, times a power of ten.

340 000 = 3.4 · 10⁵ 0.0072 = 7.2 · 10⁻³

Example

Example 1: 5² · 5³ / 5⁴

We want to simplify to a single power with base five.

  1. Same base, multiplication: we add the exponents.

    5² · 5³ = 5²⁺³ = 5⁵

  2. Division: we subtract the exponents.

    5⁵/5⁴ = 5⁵⁻⁴ = 5¹

  3. Five to the power of one is a single five.

    5¹ = 5

5² = 25, 5³ = 125 and 5⁴ = 625. 25 · 125 = 3 125 and 3 125 ÷ 625 = 5. The same answer.

Example 2: write 0.00045 in standard form

We want to write the number as a number between 1 and 10 times a power of ten.

  1. We move the decimal point four places to the right and get 4.5.

    0.00045 → 4.5

  2. That was times 10 000. We divide back by 10 000, which is times 10⁻⁴.

    0.00045 = 4.5 · 10⁻⁴

10⁻⁴ = 0.0001 and 4.5 · 0.0001 = 0.00045. We got the number back.

Pitfalls

  • Multiplying the base by the exponent: 2³ = 6.

    2³ = 2 · 2 · 2 = 8. Write out the factors when you are unsure.

  • Multiplying the exponents when multiplying powers: 2³ · 2⁴ = 2¹².

    Three twos times four twos is seven twos: 2⁷. The exponents are multiplied for a power of a power, such as (2³)⁴.

  • Thinking that 2⁻³ is negative, that is −8.

    A negative exponent gives a fraction: 2⁻³ = 1/2³ = 1/8.

Practise on your own

0 of 4 correct

  1. Work out 4³.

  2. Write 3² · 3³ as a single power and work out the value.

  3. Work out 10⁵/10⁷. Answer as a decimal.

  4. Write 6 200 000 in standard form.

More walkthroughs with Fox

Common questions

What is a number to the power of zero?

One. The pattern 2² = 4, 2¹ = 2 continues with 2⁰ = 1. This holds for every base except zero.

What is the difference between −3² and (−3)²?

In −3², the exponent belongs to the three: −(3 · 3) = −9. In (−3)², the whole of −3 is raised to the power: (−3) · (−3) = 9.

Can you add powers by adding the exponents?

No. The laws apply to multiplication and division. 2³ + 2⁴ = 8 + 16 = 24, but 2⁷ = 128.

Why should the number before the power of ten be between 1 and 10 in standard form?

Then every number is written in only one way, and the exponent shows straight away how big it is. 34 · 10⁴ = 3.4 · 10⁵, but only the second is standard form.

Keep practising with Somokai

Free to start. Hints and explanations when you get stuck.

Start practising for free

Found a mistake? Get in touch

Svenska