# Square roots and cube roots

## In a nutshell

Square roots and cube roots are the inverses of the square and cube operations.

## Square roots

The square root of a number, $x$, is denoted as $\sqrt{x}$. A square root is defined as follows:

If $x=y^2$, then $\sqrt{x}=y$.

In other words, finding the square root of $x$ is the same thing as asking *what number becomes $x$ when multiplied by itself?*

### Key facts to know about square roots

- The square root of a number is always defined to be
**positive**. - The square root of a negative number
**does not exist**.

### Finding square roots

Square roots can either by found by memorising square numbers, or by using a calculator.

##### Example 1

*What is the square root of $36$?*

*You should know that $6^2=6\times6=36$*. *Therefore, $\underline{\sqrt{36}=6}$.*

##### Example 2

*What is the square root of $85$?*

*Use the $\sqrt{}$ button on your calculator:*

$\underline{\sqrt{85}=9.219544457...}$

## Cube roots

The cube root of a number, $x$, is denoted as $\sqrt[3]{x}$. A cube root is defined as follows:

If $x=y^3$, then $\sqrt[3]{x}=y$.

In other words, finding the cube root of $x$ is the same as asking *what number, when cubed, becomes $x$?*

### Key facts to know about cube roots

- You can cube root negative numbers.

- The cube root of a negative number is negative.

### Finding cube roots

Cube numbers can be very large for even smaller numbers. So, in most cases, you will have to use your calculator to find cube roots.

##### Example 3

*What is the cube root of $512$*?

$8^3=8\times8\times8=512$, *so $\underline{\sqrt[3]{512}=8}$*.

##### Example 4

*What is the cube root of $-100$?*

*Use the $\sqrt[3]{}$ button on your calculator:*

$\underline{\sqrt[3]{-100}=-4.641588834...}$