\(\sqrt{36+16\sqrt{5}}\)
\(=\sqrt{4(9+4\sqrt{5})}\) (factor the 36 and 16 in the square root)
\(=2\sqrt{9+4\sqrt{5}}\) (take out the 4 in the square root)
\(=2\sqrt{9+2\sqrt{20}}\) (rewrote)
now let's just focus of the \(9+2\sqrt{20}\) for now
\(9+2\sqrt{20}=4+5+2\sqrt{20}\)
\(4+5+2\sqrt{20}=(\sqrt{4})^2+(\sqrt{5})^2+2\cdot\sqrt{4}\cdot\sqrt{5}\)
***you get this by using the formula: \(a^2+b^2+2ab=(a+b)^2\)***
so, \(9+2\sqrt{20}=(\sqrt{4}+{\sqrt{5}})^2\)
now we can plug that back into the original equation.
\(=2\sqrt{9+2\sqrt{20}}=2\sqrt{(\sqrt{4}+\sqrt{5})^2}\)
\(=2(\sqrt{4}+\sqrt{5})\)
\(=2(2+\sqrt{5})\)
\(=\boxed{4+2\sqrt{5}}\)
sorry if this was unclear, i was in a bit of a hurry
\(\)
.
