Question

Find the square root of the following in the form of a binomial surd.
$2+\frac{2}{3}\sqrt{5}$.

Answer 1

Let $x=\sqrt{2+\frac{2}{3}\sqrt{5}}=\sqrt{n}+\sqrt{m}$

$2+\frac{2}{3}\sqrt{5}=n+m+2\sqrt{nm}$

Hence, $n+m=2,nm=\frac{5}{9}$

$9n^2-18n+5=0$

$n=\frac{5}{3}$ or $n=\frac{1}{3}$

Similarily, $m=\frac{5}{3}$ or $m=\frac{1}{3}$