Question

If x = ${log}_{3}243,y={log}_{2}64,$, Then $\sqrt{x-2\sqrt{y}}$ is

• A. $\sqrt{5-2\sqrt{6}}$
• B. $2-\sqrt{3}$
• C. $\sqrt{3}-\sqrt{2}$
• D. $\sqrt{3}-4$

Answer 1

The correct option is A $\sqrt{5-2\sqrt{6}}$

We have,

$x={\log }_{3}243......\left(1\right)$

$y={\log }_{2}64......\left(2\right)$

Applying rule,

${\log }_{b}x=y\Rightarrow b^y=x$

So,

From equation (1) and (2) to,

$x={\log }_{3}243$

$\Rightarrow 3^x=243$

$\Rightarrow 3^x=3^5$

$\Rightarrow x=5$

Now,

${\log }_{2}64=y$

$\Rightarrow 2^y=64$

$\Rightarrow 2^y=2^6$

$\Rightarrow y=6$

Now,

$\sqrt{x-2\sqrt{y}}$

$=\sqrt{5-2\sqrt{6}}$

Hence, this is the answer.