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