# Powers and roots: Square and cube numbers

## In a nutshell

A square number is the name given to the answer when a number is multiplied by itself. A cube number is the name given when a number is multiplied by itself twice. A square root is the inverse of a square, and a cube root is the inverse of a cube. When a number is multiplied by itself more than twice, it can written using powers.

## Powers

When a number is multiplied by itself several times, it can be written using powers.

$a^m$

Where $a$ is referred to as the *base*, and $m$ is the *power *- also known as the *index *or *exponent*.

##### Example 1

*What is $2^4$?*

$2$* is multiplied by itself $4$ times.*

$2^4 = 2 \times 2 \times 2 \times 2 = \underline{16}$

## Square numbers

A square number is the product when a number is multiplied by itself. You have to know the first $15$ square numbers.

$1^2 =1 \times 1 = \underline1 \\2^2 =2 \times 2 = \underline4 \\3^2 =3 \times 3 = \underline9 \\4^2 =4 \times 4 = \underline{16} \\5^2 =5 \times 5 = \underline{25} \\6^2 =6 \times 6 = \underline{36} \\7^2 =7 \times 7 = \underline{49} \\8^2 =8 \times 8 = \underline{64} \\9^2 =9 \times 9 = \underline{81} \\10^2 =10 \times 10 = \underline{100} \\11^2 =11 \times 11 = \underline{121} \\12^2 =12 \times 12 = \underline{144} \\13^2 =13 \times 13 = \underline{169} \\14^2 =14 \times 14 = \underline{196} \\15^2 =15 \times 15 = \underline{225} \\$

## Cube numbers

A cube number is the product when a number is multiplied by itself twice. You have to know the first $5$ cube numbers.

$1^3 = 1 \times 1 \times 1 = \underline1 \\2^3 = 2 \times 2 \times 2 = \underline8 \\3^3 = 3 \times 3 \times 3 = \underline{27} \\4^3 = 4 \times 4 \times 4 = \underline{64} \\5^3 = 5 \times 5 \times 5 = \underline{125} \\$

## Square and cube roots

To find a square root, think of which number you would need to multiply itself by to get a square number. To find a cube root, think of which number was multiplied by itself twice.

##### **Example 2**

*What is $\sqrt {81}$ ?*

*Find which positive number - when multiplied by itself - gives $81$.*

*$9\times 9 = 81$*

$\underline{\sqrt{81} = 9}$

**Note***:** Square roots are defined to only be positive.*

**Example 3**

*Find the value of $\sqrt[3]{125}$.*

*Find the number that - when multiplied by itself twice - gives $125$.*

*$5 \times 5 \times 5 = 125$*

**

$\underline{\sqrt[3]{125}=5}$