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Square roots and cube roots

Tutor: Bilal

# 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:

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

## FAQs - Frequently Asked Questions

### What is the square root of a number?

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