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Powers and Roots

What you'll learn

  • How to read numbers like 424^242 and 10310^3103.
  • How to multiply the same number by itself.
  • How to work backwards from square and cube answers.
  • How to choose matching numbers from a list.

1. Small raised numbers

When you see a number like 343^434, the small raised number tells you to use repeated multiplication.

Definition

Power, base and index

A power is a short way to write repeated multiplication. In 343^434, the base is 3 and the index is 4. It means multiply four 3s together.

The base is the repeated number and the index tells how many copies to multiply.

Key Idea

Remember this

The index tells you how many copies of the base to multiply together.

Example

Working out 33

  1. Look at the base and index. In 333^333, the base is 3 and the index is 3.

  2. Write it as repeated multiplication:

    33=3×3×33^3 = 3 \times 3 \times 333=3×3×3

  3. Multiply in order:

    3×3=9,9×3=273 \times 3 = 9,\quad 9 \times 3 = 273×3=9,9×3=27

  4. So 33=273^3 = 2733=27.

Common Mistake

Do not multiply by the index

525^252 does not mean 5 multiplied by 2. It means 5 multiplied by itself: 52=5×55^2 = 5 \times 552=5×5.

Writing repeated multiplication as a power

Sometimes you are given the long multiplication and asked to write it shorter.

Example

Writing repeated multiplication as a power

  1. Suppose you have 6×6×6×66 \times 6 \times 6 \times 66×6×6×6.

Four identical factors of 6 can be written as the power 6^4.

  1. The repeated number is 6, so the base is 6.

  2. Count how many 6s there are. There are four 6s.

  3. Write this as a power:

    6×6×6×6=646 \times 6 \times 6 \times 6 = 6^46×6×6×6=64

2. Square numbers

A number is squared when it is multiplied by itself. For example, 727^272 is read as “7 squared”.

Definition

Square number

A square number is the answer you get when a number is multiplied by itself.

Squaring can be pictured as making a square array with equal side lengths.

The first few square numbers are:

  • 1, because 12=11^2 = 112=1
  • 4, because 22=42^2 = 422=4
  • 9, because 32=93^2 = 932=9
  • 16, because 42=164^2 = 1642=16
  • 25, because 52=255^2 = 2552=25
Example

Working out a square number

  1. Work out 727^272.

7^2 means a 7 by 7 square array, not 7 multiplied by 2.

  1. This means 7 multiplied by itself:

    72=7×77^2 = 7 \times 772=7×7

  2. Multiply:

    7×7=497 \times 7 = 497×7=49

  3. So 72=497^2 = 4972=49, and 49 is a square number.

Even and odd square numbers

An even number can be shared into pairs exactly, like 2, 4, 6, 8. An odd number cannot be shared into pairs exactly, like 1, 3, 5, 7.

Example

Giving an even square number

  1. Choose an even number to square, such as 6.

  2. Square it:

    62=6×6=366^2 = 6 \times 6 = 3662=6×6=36

  3. So 36 is a square number and it is even.

3. Square roots

A square root works backwards from a square number.

Definition

Square root

A square root of a number is a number that multiplies by itself to make that number. The symbol 100\sqrt{100}100​ is read as “the square root of 100”.

A square root asks for the side length of a square with a known area.

Example

Finding 100​

  1. Ask: what number multiplied by itself gives 100?

  2. Use the square fact:

    102=10×10=10010^2 = 10 \times 10 = 100102=10×10=100

  3. So 100=10\sqrt{100} = 10100​=10.

Tip

Think backwards

If 82=648^2 = 6482=64, then 64=8\sqrt{64} = 864​=8. Squaring and square rooting undo each other.

Squaring and square rooting are opposite operations.

4. Cube numbers and cube roots

A number is cubed when it is multiplied by itself three times.

Definition

Cube number and cube root

A cube number is the answer you get when a number is multiplied by itself three times. A cube root asks which number was cubed to make the answer.

Cubing can be pictured as building a cube with equal length, width and height.

Example

Working out 43

  1. Read 434^343 as “4 cubed”.

  2. Write it as repeated multiplication:

    43=4×4×44^3 = 4 \times 4 \times 443=4×4×4

  3. Multiply in stages:

    4×4=16,16×4=644 \times 4 = 16,\quad 16 \times 4 = 644×4=16,16×4=64

  4. So 43=644^3 = 6443=64.

Example

Finding the cube root of 125

  1. A cube root asks: what number cubed gives 125?

  2. Try 5 cubed:

    53=5×5×5=1255^3 = 5 \times 5 \times 5 = 12553=5×5×5=125

  3. So the cube root of 125 is 5. Written with a symbol, 1253=5\sqrt[3]{125} = 53125​=5.

Common Mistake

Cube root is not divide by 3

The cube root of 27 is 3, because 33=273^3 = 2733=27. It is not 27 divided by 3.

5. Powers of 10

Powers of 10 are especially useful because they make 1 followed by zeros.

For example:

  • 101=1010^1 = 10101=10
  • 102=10010^2 = 100102=100
  • 103=100010^3 = 1000103=1000
  • 104=1000010^4 = 10000104=10000
Tip

Counting zeros

For powers like 10310^3103 and 10410^4104, the index tells you how many zeros come after the 1.

Example

Working out 6×103

  1. Work out the power of 10 first:

    103=100010^3 = 1000103=1000

  2. Now multiply by 6:

    6×1000=60006 \times 1000 = 60006×1000=6000

  3. So 6×103=60006 \times 10^3 = 60006×103=6000.

6. Choosing numbers from a list

In list questions, you usually need to recognise a type of number.

Useful facts to learn:

  • Powers of 2: 2, 4, 8, 16, 32, 64
  • Square numbers: 1, 4, 9, 16, 25, 36, 49, 64
  • Cube numbers: 1, 8, 27, 64, 125
Definition

Power of 2

A power of 2 is made by starting with 2 and multiplying by 2 again and again.

Example

Choosing powers, squares and cubes from a list

  1. Look at this list: 4, 8, 9, 12, 16, 25, 27, 32, 36, 64.

A sorting diagram helps show which numbers are powers of 2, square numbers, cube numbers, or in more than one group.

  1. The powers of 2 in the list are 4, 8, 16, 32 and 64.

  2. The square numbers in the list are 4, 9, 16, 25, 36 and 64.

  3. The cube numbers in the list are 8, 27 and 64.

Exam technique

In the exam

  1. Read the small raised number carefully before multiplying.

  2. For roots, ask: “What number multiplied by itself, or cubed, makes this?”

  3. In list questions, write down the facts you know first, then match them to the list.

Self review

Check yourself

  • Can you explain why 626^262 is not 12?
  • What number multiplied by itself gives 81?
  • Which cube numbers can you remember up to 125?
Recap questions

1 of 5

Work out 525^252.

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In the expression 343^434, what are the names for the number 3 and the number 4?

Powers and Roots Revision Guide

  1. GCSE
  2. /Maths
  3. /Powers and Roots