Simplify the square root

[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...

[SammyS]

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...
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\,.

[ehild]

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

ehild

[songoku]

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\,.

Thanks :smile:

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

ehild

I agree :biggrin::approve: