# Powers and roots

**Square numbers**

A number multiplying by itself is called a **squaring** the number. This is written by adding a small 2 after it.

For example, the number 2 multiplied by itself can be written as

2 x 2 or 2²

The value of 2 squared is 4.

The result of any *whole* number multiplied by itself is called a **square number**. The term “to the power of two” can also be used.

**Square roots**

The **square root** is the opposite of finding the square and is shown with the following symbol:

√64 = 8

The square root of 64 is 8 and the square of 8 is 64.

Points to note:

- When two negative numbers are multiplied together, they are always positive
- The √ symbol refers to the positive square root by convention.

**The product of two square numbers**

A product of two square numbers is always another square number.

For example:

4 × 25 = 100

because

2 × 2 × 5 × 5 = 2 × 5 × 2 × 5

This can be used to help us find the square roots of larger square numbers.

**Using factors to find square roots**

A square number with factors can be used to find the square root.

For example,

**find √400**

√400 = √(4 x 100)

= √4 x √100

= 2 x 10

= 20

**find √225**

√225 = √(9 x 25)

= √9 x √25

= 3 x 5

= 15

**Finding square roots of decimals**

To find the square root of a number can be done by dividing two square numbers.

For example,

**find √0.09**

√0.09 = √(9 ÷ 100)

= √9 ÷ √100

= 3 ÷ 10

= 0.3

**find √0.0144**

√0.0144 = √(144 ÷ 10000)

= √144 ÷ √10000

= 12 ÷ 100

= 0.12

**Approximate square roots**

If a number cannot be written as a product or quotient of two square numbers then its square root cannot be found exactly. This will results in the number of digits after the decimal point being infinite and non-repeating. This is also an example of an **irrational** number.

Points to note:

- An irrational number cannot be written exactly as a fraction or decimal
- When using √2 using a calculator the result of 1.4142135622 is an approximation!

**Estimating square roots**

When estimating a square root of a number, find two square numbers which lie in between, and the result will be the square root between the result of the two square numbers.

**Negative square roots**

The root symbol refers to the positive square root by convention but negative square roots do exist.

4 × 4 = 16 **and **–4 × –4 = 16

Therefore, the square root of 16 is 4 or –4. When the symbol is used, it means the positive square root but the symbol can be written to mean both the positive and the negative square root.

**Cubes**

The following numbers are all cube numbers, 1, 8, 27, 64, and 125

**Cube numbers**

1³ = 1 x 1 x 1 = 1 ‘1 cubed’ or ‘1 to the power of 3’

2³ = 2 x 2 x 2 = 8 ‘2 cubed’ or ‘2 to the power of 3’

3³ = 2 x 2 x 2 = 27 ‘3 cubed’ or ‘3 to the power of 3’

4³ = 4 x 4 x 4 = 64 ‘4 cubed’ or ‘4 to the power of 3’

5³ = 5 x 5 x 5 = 125 ‘5 cubed’ or ‘5 to the power of 3’

**Cube roots**

Finding the cube root is the inverse of finding the cube and is written as follows:

The cube root of 125 is 5.

**Index notation**

We use index notation to show repeated multiplication by the same number.

For example:

we can use index notation to write **2 × 2 × 2 × 2 × 2** as and is also known as 2 to the power of 5.

**Using a calculator to find powers**

We can use the following key on a calculator to find powers.

For example:

to calculate the value of 7⁴ we key the following and the calculator shows 2401.