# How do you simplify 4+sqrt200 + 8sqrt49 - 2sqrt2?

Oct 9, 2015

$4 \cdot \left(15 + 2 \sqrt{2}\right)$

#### Explanation:

Your starting expression looks like this

$4 + \sqrt{200} + 8 \sqrt{49} - 2 \sqrt{2}$

The first thing to notice here is that $49$ is actually a perfect square

$49 = 7 \cdot 7 = {7}^{2}$

This means that the expression can be first simplified to

$4 + \sqrt{200} + 8 \cdot \sqrt{{7}^{2}} - 2 \sqrt{2}$

$4 + \sqrt{200} + 8 \cdot 7 - 2 \sqrt{2}$

$4 + \sqrt{200} + 56 - 2 \sqrt{2}$

$60 + \sqrt{200} - 2 \sqrt{2}$

Now take a look at $\sqrt{200}$. You know that $200$ is not a perfect square, but that you ccan write it as

$200 = 100 \cdot 2 = {10}^{2} \cdot 2$

This means that you have

$60 + \sqrt{{10}^{2} \cdot 2} - 2 \sqrt{2}$

$60 + 10 \sqrt{2} - 2 \sqrt{2}$

$60 + 8 \sqrt{2}$

Finally, use $4$ as acommon factor to get

$60 + 8 \sqrt{2} = \textcolor{g r e e n}{4 \cdot \left(15 + 2 \sqrt{2}\right)}$