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Key points

Using is a mathematical shorthand way of writing repeated multiplication using the same number.
A power is the small floating number that goes to the right of a number or letter. For example, in the expression 3², the ² is the power.

Powers activity

Play this game to see how numbers increase when using powers.

What are powers?

Using is a shorthand way of writing repeated multiplication using the same number. For example, rather than writing 4 x 4 x 4 it can be simplified to 4³. This is read as 'four to the power of three'.
This sort of notation is used when finding the area of a square or the of a cube. It can also be used to describe other calculations using repeated multiplication.

Examples

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Powers can be used as a shorthand way of writing repeated multiplication of the same digits. 3 x 3 x 3 x 3 x 3 can be written as 3⁵. This is read as 'three to the power of five'.
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7 x 7 x 7 x 7 can be written as 7⁴. This is read as 'seven to the power of four'.
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Power notation is used when finding the area of a square or the volume of a cube. 16 is a square number. It can be written as 4 x 4 or 4². This is read as '4 squared'.
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27 is a cube number. It can be written as 3 x 3 x 3 or 3³ . This is read as 3 cubed.

Using index form

A can also be known as an or an . An index, is the small floating number that goes next to a number or a letter. The plural of index is indices.
Indices show how many times a number or letter has been multiplied by itself.
Every expression has maths-specific language to describe each part. For example, 2⁷ is written in index form:
- The 2 (larger digit) is called the .
- The 7 (smaller digit) is called the number. Sometimes this is called the or the .

Examples

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4⁶ is an example of an expression that has been written using index form. This is read as 'four to the power of six'.
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The expression is made from two parts. The 4 (larger digit) is the base number. The 6 (smaller digit) is the index number (or power/exponent).
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The number that is repeatedly multiplied is the 4 (the base number). The index of 6 means that 4 is used 6 times in the multiplication. 4 is the base and 6 is the index number. 4⁶ = 4 x 4 x 4 x 4 x 4 x 4
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The expression is made from two parts. The 5 (larger digit) is the base number. The 3 (smaller digit) is the index number (or power/exponent). This is read as 'five cubed'.
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The number that is repeatedly multiplied is the 5 (the base number). The index of 3 means that 5 is used 3 times in the multiplication. 5 is the base and 3 is the index number. 5³ = 5 x 5 x 5
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Indices and powers are always written using the same type of notation. The base number can be represented by a number or a letter. The ‘a’ (larger digit) is the base number. The 4 (smaller digit) is the index number (or power/exponent). This is read as ‘a to the power of four'.
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The value that is repeatedly multiplied is a (the base number). The index of 4 means that ‘a’ is used 4 times in the multiplication. ‘a’ is the base and 4 is the index number. a⁴ = a x a x a x a

Using a calculator

When asked to calculate or work out the value of an expression that is written using power/index notation, some can be completed mentally, whilst other more complicated calculations would involve using a scientific calculator.
All scientific calculators have a ‘power’ button. This is particularly useful when the index number is large.

Examples

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Work out the value of 5 cubed.
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Interpret the calculation using the base and the index number. 5 cubed means 5 x 5 x 5. A question like this can be solved without using a calculator.
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5 is the base. 3 is the index (or power/exponent). 5³ = 5 x 5 x 5. 5 x 5 x 5 = 125
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Work out the value of 12 to the power of five (12⁵). A question like this should be solved using a calculator.
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Interpret the calculation using the base and the index number. Using a basic calculator type '12 x 12 x 12 x 12 x 12 ='. The answer should be 248,832
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If using a scientific calculator to work out 12⁵, identify the power button. These appear differently, depending on the calculator.
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To work out 12⁵ on a scientific calculator: type in the base (12), press the power button, enter the index (or power/exponent) (5), then press =
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Using a calculator, 12⁵ = 248,832. 12 x 12 x 12 x 12 x 12 = 248,832

Question

Work out the value of 6 cubed.

Six cubed equals a highlighted question mark.

Practise powers

Practise powers in this quiz.

Quiz

Real-world maths

Man painting a wall with a roller — Image caption,
A decorator might calculate the area of a room to be painted.

Using powers is a strategy that is used in everyday life to help solve problems.

A painter or decorator may use powers to calculate and record the area of a square room. For example, a room in a house may measure 4.8 m x 4.2 m, this is an area of 20.16 square metres, this is written as 20.16 m².

Man painting a wall with a roller — Image caption,
A decorator might calculate the area of a room to be painted.

Game - Divided Islands

More on Powers and roots

Find out more by working through a topic

Roots
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2 of 3
Laws of indices for multiplication and division
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3 of 3