Question

Problems Set at an Oral Examination.
Simplify the following expressions.
$\sqrt{a}\sqrt[4]{4a}\sqrt[8]{8a}...\sqrt[512]{512a}.$

Answer 1

$\sqrt{a}\sqrt[4]{4a}\sqrt[8]{8a}...\sqrt[512]{512a}$

$=a^{(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{512})}$

Now-

$\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{512}$

$=\frac{1}{2}+\frac{1}{2^2}+\frac{1}{2^3}+...+\frac{1}{2^9}$

$=\frac{\frac{1}{2}(\frac{1}{2^9}-1)}{\frac{1}{2}-1}$

$=1-\frac{1}{2^9}=\frac{511}{512}$

Hence we get-

$a^{\frac{511}{512}}$