#### Which of the following is equal to x?(A)$x^{\frac{12}{7}}+x^{\frac{5}{7}}$(B)$\sqrt[12]{\left ( x^{4} \right )^{\frac{1}{3}}}$(C)$\left ( \sqrt{x^{3}} \right )^{\frac{2}{3}}$(D)$x^{\frac{12}{7}}\times x^{\frac{7}{12}}$

Solution.
(A) We have,

$x^{\frac{12}{7}}+x^{\frac{5}{7}}= x^{\frac{1}{7}\left ( 12 \right )}+x^{\frac{1}{7}\left ( 5 \right )}$
$= x^{\frac{1}{7}}\left ( x^{12} +x^{5}\right )\neq x$
(B) We have,

$\sqrt[12]{\left ( x^{4} \right )^{\frac{1}{3}}}=\left ( \left ( x^{4} \right ) ^{\frac{1}{3}}\right )^{\frac{1}{12}}$ $\left ( \sqrt[n]{a} = \left ( a \right )^{\frac{1}{n}}\right )$

$= x^{4\times ^{\frac{1}{3}\times \frac{1}{12}}}= x^{\frac{1}{9}}$             $\because \left ( a^{m} \right )^{n}= a^{m\times n}$

$\neq x$

(C) We have,

$\left ( \sqrt{x^{3}} \right )^{\frac{2}{3}}= \left ( \left ( x^{3} \right ) ^{\frac{2}{3}}\right )^{\frac{1}{2}}$       $\left ( \sqrt[n]{a} = \left ( a \right )^{\frac{1}{n}}\right )$

$= x^{3\times \frac{2}{3}\times \frac{1}{2}}$         $\because \left ( a^{m} \right )^{n}= a^{m\times n}$

= x

(D) We have,

$x^{\frac{12}{7}}\times x^{\frac{7}{12}}= x^{\frac{12}{7}+\frac{7}{12}}= x^{\frac{144+49}{84}}\neq x$
$\because a^{m}\times a^{n}= a^{m+n}$

Hence option C is correct.