How to Simplify a Square Root with Multiple Radicands

[songoku]

Homework Statement

Simplify

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

Homework Equations

\sqrt{a+b+2\sqrt{ab}} = \sqrt{a}+\sqrt{b}

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]

Homework Statement

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

Homework Equations

\sqrt{a+b+2\sqrt{ab}} = \sqrt{a}+\sqrt{b}

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!

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

SammyS, you are the greatest square-root simplifier!

ehild

I agree