Powers and Roots

What you'll learn

What powers mean, including the words base and index
How to work out simple squares, cubes and powers of 10
How square roots and cube roots “undo” powers
How to spot square numbers, cube numbers and powers of 2 in a list

Start point: repeated multiplication

Before powers, make sure you are happy with repeated multiplication. This means multiplying by the same number again and again.

For example, $\times 6 \times 6$ means “6 multiplied by 6, then multiplied by 6 again”.

Definition

Factor

A factor is one of the numbers being multiplied. In $\times 6 \times 6$ , each 6 is a factor.

Powers: a short way to write repeated multiplication

A power is a shortcut for repeated multiplication.

Definition

Power, base and index

In $a^n$ , the number $a$ is the base and the number $n$ is the index. The index tells you how many times to use the base as a factor.

So $4^3$ means:

\times 4 \times 4

You say $4^3$ as “4 to the power of 3”, or sometimes “4 cubed”.

Example

Writing repeated multiplication as a power

Write $\times 7 \times 7 \times 7$ as a power.

Look for the repeated factor. The same factor is 7.
Count how many times 7 appears. It appears 4 times.
Use 7 as the base and 4 as the index.
Write the answer as:
$7^4$

Example

Working out a power

Work out $2^4$ .

The base is 2 and the index is 4.
Write it as repeated multiplication.
$24=2×2×2×22^4 = 2 \times 2 \times 2 \times 2$
Multiply from left to right: $\times 2 = 4$ , then $\times 2 = 8$ , then $\times 2 = 16$ .
The value is:
$2^4 = 16$

Common Mistake

Do not multiply the base by the index

$3^2$ means $\times 3$ , not $\times 2$ . So $3^2=9$ , not 6.

Squares and square numbers

When the index is 2, we say the number is squared.

Definition

Square number

A square number is the answer you get when a whole number is multiplied by itself. For example, $5^2=25$ , so 25 is a square number.

The word “square” comes from arranging objects in equal rows and columns, like a square grid.

A labelled diagram showing a 3 by 3 square array for 3 squared and a 2 by 2 by 2 cube for 2 cubed

The first few square numbers are:

1, because $1^2=1$
4, because $2^2=4$
9, because $3^2=9$
16, because $4^2=16$
25, because $5^2=25$
36, 49, 64, 81, 100

Example

Working out a square

Find the value of $8^2$ .

Squared means “multiply the number by itself”.
Write $8^2$ as a multiplication.
$82=8×88^2 = 8 \times 8$
Work out the multiplication.
$\times 8 = 64$

Example

Finding a square number with a property

Write down an even square number.

Choose a whole number to square. To make the answer even, choose an even number such as 4.
Square the number.
$42=4×4=164^2 = 4 \times 4 = 16$
16 is an even square number.

Tip

Useful squares to learn

Try to know the square numbers from 1 to 100. They come up often: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.

Square roots

A root goes backwards from a power. A square root goes backwards from squaring.

Definition

Square root

The square root of a number is the positive number that squares to make it. The symbol $n\sqrt{n}$ means “the square root of $n$ ”.

For example, $49=7\sqrt{49}=7$ because $7^2=49$ .

Example

Finding a square root

Find $81\sqrt{81}$ .

Ask: “Which number squared gives 81?”
Use your square number facts.
$9^2 = 81$
Therefore:
$81=9\sqrt{81} = 9$

Key Idea

Squares and square roots undo each other

If $6^2=36$ , then $36=6\sqrt{36}=6$ . Squaring moves one way; square rooting moves back.

Cubes and cube numbers

When the index is 3, we say the number is cubed.

Definition

Cube number

A cube number is the answer you get when a whole number is multiplied by itself three times. For example, $4^3=64$ , so 64 is a cube number.

The first few cube numbers are:

1, because $1^3=1$
8, because $2^3=8$
27, because $3^3=27$
64, because $4^3=64$
125, because $5^3=125$

Example

Working out a cube

Work out $5^3$ .

The base is 5 and the index is 3.
Write it as repeated multiplication.
$53=5×5×55^3 = 5 \times 5 \times 5$
Multiply carefully: $\times 5 = 25$ , then $25 \times 5 = 125$ .
So:
$5^3 = 125$

Cube roots

A cube root goes backwards from cubing.

Definition

Cube root

The cube root of a number is the number that cubes to make it. The cube root of 64 is 4 because $4^3=64$ .

You may also see cube root written using the symbol $n3\sqrt[3]{n}$ , but questions at this level often write it in words.

Example

Finding a cube root

Find the cube root of 27.

Ask: “Which number cubed gives 27?”
Check small cube numbers.
$33=3×3×3=273^3 = 3 \times 3 \times 3 = 27$
The cube root of 27 is 3.

Powers of 10

Powers of 10 are very common because our number system is based on tens.

The index tells you how many zeros come after the 1:

101=10102=100103=1000104=10000\begin{aligned} 10^1 &= 10\\ 10^2 &= 100\\ 10^3 &= 1000\\ 10^4 &= 10000 \end{aligned}

Example

Multiplying by a power of 10

Work out $\times 10^3$ .

Work out the power of 10 first.
$10^3 = 1000$
Replace $10^3$ with 1000.
$\times 10^3 = 6 \times 1000$
Multiply by 1000.
$\times 1000 = 6000$

Tip

Powers first

In an expression like $\times 10^3$ , calculate $10^3$ first, then multiply by 6.

Spotting powers and roots in a list

Sometimes you are given a list and asked to pick out special numbers.

For Grade 1, the most useful lists to recognise are:

powers of 2: 2, 4, 8, 16, 32, 64
square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
cube numbers: 1, 8, 27, 64, 125

Example

Choosing powers of 2 from a list

From the numbers 3, 4, 7, 8, 12, 16, 18 and 25, choose the powers of 2.

Recall the powers of 2.
$21=2,22=4,23=8,24=16,25=322^1=2,\quad 2^2=4,\quad 2^3=8,\quad 2^4=16,\quad 2^5=32$
Compare the given list with these values.
The matching numbers are 4, 8 and 16.

Example

Choosing square and cube numbers from a list

From the numbers 6, 9, 15, 16, 25, 27, 32 and 64, choose the square numbers and cube numbers.

Compare with the square numbers: 1, 4, 9, 16, 25, 36, 49, 64.
The square numbers in the list are 9, 16, 25 and 64.
Compare with the cube numbers: 1, 8, 27, 64, 125.
The cube numbers in the list are 27 and 64.

Exam technique

In the exam

Read the wording carefully: “square”, “square root”, “cube” and “cube root” all mean different things.
For powers, expand the expression if you are unsure: $5^3$ means $\times 5 \times 5$ .
For list questions, write down the known square numbers, cube numbers or powers of 2 first, then tick off the matches.

Self review

Check yourself

Can you explain the difference between $4^2$ and $2^4$ ?
Which square numbers up to 100 can you remember without working out?
How would you find the cube root of a number like 64?

Powers and Roots

What you'll learn

What powers mean, including the words base and index
How to work out simple squares, cubes and powers of 10
How square roots and cube roots “undo” powers
How to spot square numbers, cube numbers and powers of 2 in a list

Start point: repeated multiplication

Before powers, make sure you are happy with repeated multiplication. This means multiplying by the same number again and again.

For example, $\times 6 \times 6$ means “6 multiplied by 6, then multiplied by 6 again”.

Definition

Factor

A factor is one of the numbers being multiplied. In $\times 6 \times 6$ , each 6 is a factor.

Powers: a short way to write repeated multiplication

A power is a shortcut for repeated multiplication.

Definition

Power, base and index

In $a^n$ , the number $a$ is the base and the number $n$ is the index. The index tells you how many times to use the base as a factor.

So $4^3$ means:

\times 4 \times 4

You say $4^3$ as “4 to the power of 3”, or sometimes “4 cubed”.

Example

Writing repeated multiplication as a power

Write $\times 7 \times 7 \times 7$ as a power.

Look for the repeated factor. The same factor is 7.
Count how many times 7 appears. It appears 4 times.
Use 7 as the base and 4 as the index.
Write the answer as:
$7^4$

Example

Working out a power

Work out $2^4$ .

The base is 2 and the index is 4.
Write it as repeated multiplication.
$24=2×2×2×22^4 = 2 \times 2 \times 2 \times 2$
Multiply from left to right: $\times 2 = 4$ , then $\times 2 = 8$ , then $\times 2 = 16$ .
The value is:
$2^4 = 16$

Common Mistake

Do not multiply the base by the index

$3^2$ means $\times 3$ , not $\times 2$ . So $3^2=9$ , not 6.

Squares and square numbers

When the index is 2, we say the number is squared.

Definition

Square number

A square number is the answer you get when a whole number is multiplied by itself. For example, $5^2=25$ , so 25 is a square number.

The word “square” comes from arranging objects in equal rows and columns, like a square grid.

A labelled diagram showing a 3 by 3 square array for 3 squared and a 2 by 2 by 2 cube for 2 cubed

The first few square numbers are:

1, because $1^2=1$
4, because $2^2=4$
9, because $3^2=9$
16, because $4^2=16$
25, because $5^2=25$
36, 49, 64, 81, 100

Example

Working out a square

Find the value of $8^2$ .

Squared means “multiply the number by itself”.
Write $8^2$ as a multiplication.
$82=8×88^2 = 8 \times 8$
Work out the multiplication.
$\times 8 = 64$

Example

Finding a square number with a property

Write down an even square number.

Choose a whole number to square. To make the answer even, choose an even number such as 4.
Square the number.
$42=4×4=164^2 = 4 \times 4 = 16$
16 is an even square number.

Tip

Useful squares to learn

Try to know the square numbers from 1 to 100. They come up often: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100.

Square roots

A root goes backwards from a power. A square root goes backwards from squaring.

Definition

Square root

The square root of a number is the positive number that squares to make it. The symbol $n\sqrt{n}$ means “the square root of $n$ ”.

For example, $49=7\sqrt{49}=7$ because $7^2=49$ .

Example

Finding a square root

Find $81\sqrt{81}$ .

Ask: “Which number squared gives 81?”
Use your square number facts.
$9^2 = 81$
Therefore:
$81=9\sqrt{81} = 9$

Key Idea

Squares and square roots undo each other

If $6^2=36$ , then $36=6\sqrt{36}=6$ . Squaring moves one way; square rooting moves back.

Cubes and cube numbers

When the index is 3, we say the number is cubed.

Definition

Cube number

A cube number is the answer you get when a whole number is multiplied by itself three times. For example, $4^3=64$ , so 64 is a cube number.

The first few cube numbers are:

1, because $1^3=1$
8, because $2^3=8$
27, because $3^3=27$
64, because $4^3=64$
125, because $5^3=125$

Example

Working out a cube

Work out $5^3$ .

The base is 5 and the index is 3.
Write it as repeated multiplication.
$53=5×5×55^3 = 5 \times 5 \times 5$
Multiply carefully: $\times 5 = 25$ , then $25 \times 5 = 125$ .
So:
$5^3 = 125$

Cube roots

A cube root goes backwards from cubing.

Definition

Cube root

The cube root of a number is the number that cubes to make it. The cube root of 64 is 4 because $4^3=64$ .

You may also see cube root written using the symbol $n3\sqrt[3]{n}$ , but questions at this level often write it in words.

Example

Finding a cube root

Find the cube root of 27.

Ask: “Which number cubed gives 27?”
Check small cube numbers.
$33=3×3×3=273^3 = 3 \times 3 \times 3 = 27$
The cube root of 27 is 3.

Powers of 10

Powers of 10 are very common because our number system is based on tens.

The index tells you how many zeros come after the 1:

101=10102=100103=1000104=10000\begin{aligned} 10^1 &= 10\\ 10^2 &= 100\\ 10^3 &= 1000\\ 10^4 &= 10000 \end{aligned}

Example

Multiplying by a power of 10

Work out $\times 10^3$ .

Work out the power of 10 first.
$10^3 = 1000$
Replace $10^3$ with 1000.
$\times 10^3 = 6 \times 1000$
Multiply by 1000.
$\times 1000 = 6000$

Tip

Powers first

In an expression like $\times 10^3$ , calculate $10^3$ first, then multiply by 6.

Spotting powers and roots in a list

Sometimes you are given a list and asked to pick out special numbers.

For Grade 1, the most useful lists to recognise are:

powers of 2: 2, 4, 8, 16, 32, 64
square numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100
cube numbers: 1, 8, 27, 64, 125

Example

Choosing powers of 2 from a list

From the numbers 3, 4, 7, 8, 12, 16, 18 and 25, choose the powers of 2.

Recall the powers of 2.
$21=2,22=4,23=8,24=16,25=322^1=2,\quad 2^2=4,\quad 2^3=8,\quad 2^4=16,\quad 2^5=32$
Compare the given list with these values.
The matching numbers are 4, 8 and 16.

Example

Choosing square and cube numbers from a list

From the numbers 6, 9, 15, 16, 25, 27, 32 and 64, choose the square numbers and cube numbers.

Compare with the square numbers: 1, 4, 9, 16, 25, 36, 49, 64.
The square numbers in the list are 9, 16, 25 and 64.
Compare with the cube numbers: 1, 8, 27, 64, 125.
The cube numbers in the list are 27 and 64.

Exam technique

In the exam

Read the wording carefully: “square”, “square root”, “cube” and “cube root” all mean different things.
For powers, expand the expression if you are unsure: $5^3$ means $\times 5 \times 5$ .
For list questions, write down the known square numbers, cube numbers or powers of 2 first, then tick off the matches.

Self review

Check yourself

Can you explain the difference between $4^2$ and $2^4$ ?
Which square numbers up to 100 can you remember without working out?
How would you find the cube root of a number like 64?

Powers and Roots

What you'll learn

Start point: repeated multiplication

Powers: a short way to write repeated multiplication

Squares and square numbers

Square roots

Cubes and cube numbers

Cube roots

Powers of 10

Spotting powers and roots in a list

Recap questions

You've reached the end

Powers and Roots

What you'll learn

Start point: repeated multiplication

Powers: a short way to write repeated multiplication

Squares and square numbers

Square roots

Cubes and cube numbers

Cube roots

Powers of 10

Spotting powers and roots in a list

Recap questions

You've reached the end

Powers and Roots

What you'll learn

Start point: repeated multiplication

Powers: a short way to write repeated multiplication

Squares and square numbers

Square roots

Cubes and cube numbers

Cube roots

Powers of 10

Spotting powers and roots in a list

Recap questions

You've reached the end

Powers and Roots Revision Guide

Powers and Roots

What you'll learn

Start point: repeated multiplication

Powers: a short way to write repeated multiplication

Squares and square numbers

Square roots

Cubes and cube numbers

Cube roots

Powers of 10

Spotting powers and roots in a list

Recap questions

You've reached the end

Powers and Roots Revision Guide