Question

The value of $\sqrt{\left(\sqrt[2010]{20102\sqrt{7}-3\sqrt{3}}\right)\left(\sqrt[4020]{402055+12\sqrt{21}}\right)}$ is

• A. 0
• B. 10
• C. 1
• D. 4

Answer 1

The correct option is C 1

$\sqrt{{(2\sqrt{7}-3\sqrt{3})}^{\frac{1}{2010}}{(55+12\sqrt{21})}^{\frac{1}{4020}}}$
$=\sqrt{{(2\sqrt{7}-3\sqrt{3})}^{\frac{1}{2010}}{({\left(2\sqrt{7}\right)}^2+{\left(3\sqrt{3}\right)}^2+2\times 2\sqrt{7}\times 3\sqrt{3})}^{\frac{1}{4020}}}$
$=\sqrt{{(2\sqrt{7}-3\sqrt{3})}^{\frac{1}{2010}}{(2\sqrt{7}+3\sqrt{3})}^{\frac{2}{4020}}}$
$=\sqrt{{(2\sqrt{7}-3\sqrt{3})}^{\frac{1}{2010}}{(2\sqrt{7}+3\sqrt{3})}^{\frac{1}{2010}}}$
$=\sqrt{{\left[(2\sqrt{7}-3\sqrt{3})(2\sqrt{7}+3\sqrt{3})\right]}^{\frac{1}{2010}}}$
$=\sqrt{{(28-27)}^{\frac{1}{2010}}}$
$={\left(1\right)}^{\frac{1}{4020}}=1$
Hence, the correct answer is option (c).