# What is the square root of 18 - the square root of 8?

May 3, 2018

$\sqrt{2}$

#### Explanation:

These can't be subtracted from each other because they are not like terms. So let's simplify the problem to fix that.

$\sqrt{18} - \sqrt{8}$

First break down each of the squares into prime numbers:

$\sqrt{3 \cdot 3 \cdot 2} - \sqrt{2 \cdot 2 \cdot 2}$

Now combine any factors that make perfect squares that we can factor out:

$\sqrt{{3}^{2} \cdot 2} - \sqrt{{2}^{2} \cdot 2}$

Now factor those squares out of the radical sign. Because they are able to be squared perfectly, we can take them out of the radical. But the other terms will be left behind because they cannot be squared perfectly:

$3 \sqrt{2} - 2 \sqrt{2}$

Now subtract as normal, since they are both being multiplied by $\sqrt{2}$, they are like terms and can be subtracted:

$3 \sqrt{2} - 2 \sqrt{2} = \textcolor{red}{1 \sqrt{2}}$

$1 \sqrt{2} \rightarrow \sqrt{2}$

Anything multiplied by $1$ is itself, so $1 \sqrt{2}$ is the same thing as $\sqrt{2}$. That is the simplest form of this subtraction problem.