# Simplify the square root

1. Jan 19, 2012

### songoku

1. The problem statement, all variables and given/known data
Simplify

$$\sqrt {10 + \sqrt{24} + \sqrt{40}+\sqrt{60}}$$

2. Relevant equations
$$\sqrt{a+b+2\sqrt{ab}} = \sqrt{a}+\sqrt{b}$$

3. The attempt at a solution
$$\sqrt {10 + \sqrt{24} + \sqrt{40}+\sqrt{60}}$$
$$= \sqrt {10 + 2 \sqrt{6} + 2 \sqrt{10}+ 2 \sqrt{15}}$$

Stuck...

2. Jan 19, 2012

### SammyS

Staff Emeritus
Why is it that $\sqrt{a+b+2\sqrt{ab}} = \sqrt{a}+\sqrt{b}\,?$

It's because

$a+b+2\sqrt{ab}=\sqrt{a}^2+2\sqrt{a}\sqrt{b}+\sqrt{b}^2$
$\displaystyle=\left(\sqrt{a}+\sqrt{b}\right)^2$​

Now look at your final expression: $\sqrt {10 + 2 \sqrt{6} + 2 \sqrt{10}+ 2 \sqrt{15}}\,.$

What are the prime factors of 6? ... of 10? ... of 15 ?

Expand $(x+y+z)^2\,.$

3. Jan 19, 2012

### ehild

SammyS, you are the greatest square-root simplifier!!!

ehild

4. Jan 19, 2012

Thanks

I agree