Question

Find the fourth root of $7+4\sqrt{3}$.

Answer 1

Let $y=(7+4\sqrt{3}{)}^{1/4}$

$7+4\sqrt{3}=3+4+2\left(2\right)\left(\sqrt{3}\right)=(\sqrt{3}{)}^2+(2{)}^2+2\left(\sqrt{3}\right)\left(2\right)$

$=(\sqrt{3}+2{)}^2$ $=\frac{1}{4}{(2\sqrt{3}+4)}^2=\frac{1}{4}\left[\right(\sqrt{3}{)}^2+(1{)}^2+2(\sqrt{3}\left)\right(1\left)\right]$

$=\frac{1}{4}(\sqrt{3}+1{)}^4$

$\Rightarrow y=\frac{\sqrt{3}+1}{4^{1/4}}=\frac{\sqrt{3}+1}{\sqrt{2}}$.