How to Simplify Radical Expressions with Multiple Radicals?

[Albert1]

simplify:

$\sqrt {21-4 \sqrt 5 +8\sqrt 3 - 4\sqrt {15}}$

 

[MarkFL]

Here is my solution:

Spoiler

$$21+8\sqrt{3}-4\sqrt{5}-4\sqrt{15}=$$

$$4+4\sqrt{3}-2\sqrt{5}+4\sqrt{3}+12-2\sqrt{15}-2\sqrt{5}-2\sqrt{15}+5=$$

$$\left(2+2\sqrt{3}-\sqrt{5} \right)^2$$

Hence:

$$\sqrt{21+8\sqrt{3}-4\sqrt{5}-4\sqrt{15}}=2+2\sqrt{3}-\sqrt{5}$$

 

[Albert1]

Here is my solution:

Spoiler

$$21+8\sqrt{3}-4\sqrt{5}-4\sqrt{15}=$$

$$4+4\sqrt{3}-2\sqrt{5}+4\sqrt{3}+12-2\sqrt{15}-2\sqrt{5}-2\sqrt{15}+5=$$

$$\left(2+2\sqrt{3}-\sqrt{5} \right)^2$$

Hence:

$$\sqrt{21+8\sqrt{3}-4\sqrt{5}-4\sqrt{15}}=2+2\sqrt{3}-\sqrt{5}$$

good solution :)