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# Roots

### Roots

The root of a number is a number which, when multiplied by itself, will make the original number.

Roots reverse the process of indices, to find the original number.

Imagine a number, let's call it $x$. If we multiply this by itself we get $x^2$. This is the square number.

$x$ would be the **square root** as it is the number that was originally squared.

What is a square root?

Let's think about this concept with numbers.

The square root of $25$ is $5$ because we know that $5 \times 5 = 25$

We can denote this with a $\sqrt{}$ sign:

$\sqrt{25}=5$

How would you write the following phrase? "Square root of 16"

Multiplying two negatives together gives a positive number. Therefore,$25$ has another root:

$-5 \times -5=25$

We can describe the positive and negative square roots with a $\pm$ sign - this indicates both a positive and a negative option.

$\sqrt{25}= \pm5$

A positive number always has two square roots

A positive and a negative

Find the values of $\sqrt{36}$

Work out the values of $\sqrt{100}$

A **cube root** of a number is a number which, when multiplied by itself 3 times, generates the original number. The cube root of 64 is 4.

$4 \times 4\times4=64$

We indicate a cube root with the $\sqrt[3]{}$symbol.

$\sqrt[3]{64}=4$

How would you write "the cube root of 125"?

Unlike square roots, a number will only have **one** cube root, but we can consider positive and negative numbers. For example:

$\sqrt[3]{-8}=-2$

A number only has one cube root

But it can be either POSITIVE or NEGATIVE

What is the cube root of $27$?

Work out the value of $\sqrt[3]{125}$.

The square root of 0 is 0. This is because: $0 \times 0 = 0$