Question

Given that $x=6-4\sqrt{2}$ and $y$ is conjugate of $x,$ the value of $\sqrt{4xy}$ is .

Answer 1

Given,
$x=6-4\sqrt{2}$
Conjugate of $x=6+4\sqrt{2}=y$

Substituting these values in the required expression,
$\sqrt{4xy}$
$=\sqrt{4\times (6-4\sqrt{2})\times (6+4\sqrt{2})}$
(using $(a-b)(a+b)=a^2+b^2$)
$=\sqrt{4\times (6^2-(4\sqrt{2}{)}^2)}$
$=\sqrt{4\times (36-16\times 2)}$
$=\sqrt{4\times (36-32)}$
$=\sqrt{4\times 4}$
$=4$