Simplify Radical Expression

[Albert]

simplify:

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

 

[MarkFL]

Here is my solution:

Spoiler

\(\displaystyle 21+8\sqrt{3}-4\sqrt{5}-4\sqrt{15}=\)

\(\displaystyle 4+4\sqrt{3}-2\sqrt{5}+4\sqrt{3}+12-2\sqrt{15}-2\sqrt{5}-2\sqrt{15}+5=\)

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

Hence:

\(\displaystyle \sqrt{21+8\sqrt{3}-4\sqrt{5}-4\sqrt{15}}=2+2\sqrt{3}-\sqrt{5}\)

 

[Albert]

Here is my solution:

Spoiler

\(\displaystyle 21+8\sqrt{3}-4\sqrt{5}-4\sqrt{15}=\)

\(\displaystyle 4+4\sqrt{3}-2\sqrt{5}+4\sqrt{3}+12-2\sqrt{15}-2\sqrt{5}-2\sqrt{15}+5=\)

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

Hence:

\(\displaystyle \sqrt{21+8\sqrt{3}-4\sqrt{5}-4\sqrt{15}}=2+2\sqrt{3}-\sqrt{5}\)

good solution