# How do you simplify the expression 2sqrt(1/2)+2sqrt2-sqrt8?

May 7, 2017

2sqrt(1/2)+2sqrt2-sqrt8=color(blue)(sqrt2

#### Explanation:

Simplify.

$2 \sqrt{\frac{1}{2}} + 2 \sqrt{2} - \sqrt{8}$

In order to add or subtract numbers with square roots, the square roots must be the same.

Simplify $\sqrt{8}$ by prime factorization.

$2 \sqrt{\frac{1}{2}} + 2 \sqrt{2} - \sqrt{2 \times 2 \times 2}$

$2 \sqrt{\frac{1}{2}} + 2 \sqrt{2} - \sqrt{{2}^{2} \times 2}$

$2 \sqrt{\frac{1}{2}} + 2 \sqrt{2} - 2 \sqrt{2}$

Simplify $\sqrt{\frac{1}{2}}$ to $\frac{\sqrt{1}}{\sqrt{2}}$.

$2 \times \frac{\sqrt{1}}{\sqrt{2}} + 2 \sqrt{2} - 2 \sqrt{2}$

Simplify $\sqrt{1}$ to $1$.

$2 \times \frac{1}{\sqrt{2}} + 2 \sqrt{2} - 2 \sqrt{2}$

Rationalize the denominator by multiplying the numerator and denominator by color(red)(sqrt2.

$2 \times \frac{1}{\sqrt{2}} \times \frac{\textcolor{red}{\sqrt{2}}}{\textcolor{red}{\sqrt{2}}} + 2 \sqrt{2} - 2 \sqrt{2}$

Simplify.

$\frac{2 \times \sqrt{2}}{\sqrt{2} \times \sqrt{2}} + 2 \sqrt{2} - 2 \sqrt{2}$

Simplify.

$\frac{2 \sqrt{2}}{2} + 2 \sqrt{2} - 2 \sqrt{2}$

Cancel the $\frac{2}{2}$.

$\frac{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}} \sqrt{2}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}}} + 2 \sqrt{2} - 2 \sqrt{2}$

$\sqrt{2} + 2 \sqrt{2} - 2 \sqrt{2}$

Simplify.

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

$\sqrt{2}$