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Which of the following is equal to x?


(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}}

Answers (1)

Answer. [C]
(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.


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