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Powers

\({9}\) is a square number.

\(3 \times 3 = 9\)

\(3 \times 3\) can also be written as \(3^2\).

This is pronounced "\({3}\) squared".

\({8}\) is a cube number.

\(2 \times 2 \times 2 = 8\)

\(2 \times 2 \times 2\) can also be written as \(2^3\), which is pronounced "\({2}\) cubed".

Index form

The notation \(3^2\) and \(2^3\) is known as index form.

The small digit is called the index number or power.

You have already seen that \(3^2 = 3 \times 3 = 9\) and that \(2^3 = 2 \times 2 \times 2 = 8\).

Similarly, \(5^4\) (five to the power of \({4}\)) \(= 5 \times 5 \times 5 \times 5 = 625\) and \(3^5\) (three to the power of \({5}\)) \(= 3 \times 3 \times 3 \times 3 \times 3 = 243\).

The index number tells you how many times the number should be multiplied.

• When the index number is two, the number has been squared.
• When the index number is three, the number has been cubed.
• When the index number is greater than three you say that it has been multiplied to the power of.

For example:

\(7^2\) is 'seven squared'.\(3^3\) is 'three cubed'.\(3^7\) is 'three to the power of seven'.\(4^5\) is 'four to the power of five'.

All scientific calculators have a 'power' button.

This button may be labelled \(x^y\) or \(y^x\) or \({x}\)^■.

Check to find the ‘power’ button on your own calculator.

This is particularly useful when the index number is large.

Example

To work out \(4^{10}\):

• enter \({4}\)
• press the power button
• enter \({10}\)
• press \({=}\)

You should get the answer \({1,048,576}\).

Square root

The opposite of squaring a number is called finding the square root.

The symbol for the square root is \(\sqrt{}\).

Example

The square root of \({16}\) is \({4}\) (because \(4^2 = 4 \times 4 = 16\)).

The square root of \({25}\) is \({5}\) (because \(5^2 = 5 \times 5 = 25\)).

The square root of \({100}\) is \({10}\) (because \(10^2 = 10 \times 10 = 100\)).

The symbol \(\sqrt{}\) means square root, so:

\(\sqrt{36}\) means 'the square root of \({36}\)'.

\(\sqrt{36} = 6\)

\(\sqrt{81}\) means 'the square root of \({81}\)'.

\(\sqrt{81} = 9\)

You will also find a square root key on your calculator.

Cube root

The opposite of cubing a number is called finding the cube root.

The symbol for the cube root is \(^{3}\sqrt{}\).

Example

The cube root of \({27}\) is \({3}\) (because \(3 \times 3 \times 3 = 27\)).

The cube root of \({1,000}\) is \({10}\) (because \(10 \times 10 \times 10 = 1,000\)).

Example

The symbol \(^{3}\sqrt{}\) means cube root, so:

\(^{3}\sqrt{125}\), means the cube root of \({125}\).

\(^{3}\sqrt{125}=5\)

\(^{3}\sqrt{64}\) means the cube root of \({64}\).

\(^{3}\sqrt{64}=4\)

How can we work out \(2^3 \times 2^5\)?

\(2^3 = 2 \times 2 \times 2\)

\(2^5 = 2 \times 2 \times 2 \times 2 \times 2\) so \(2^3 \times 2^5 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^8\)

There are \({3}\) twos from \(2^3\) and \({5}\) twos from \(2^5\), so altogether there are \({8}\) twos.

The index \({8}\) can be found from adding the indices \({3}\) and \({5}\).

Remember, this only works when we are multiplying powers of the same number together.

In general: \(2^m \times 2^n =2^{(m + n)}\)

Examples

\(2^5 \times 2^4 = 2^{(5 + 4)} = 2^9\)

\(2^7 \times 2^3 = 2^{(7 + 3)} = 2^{10}\)

The rule also works for other numbers, so:

\(3^4 \times 3^2 = 3^{(4 + 2)} = 3^6\)

\(15^6 \times 15^4 = 15^{(6 + 4)} = 15^{10}\)