Question

how we can solve this question ​

Answer 1

Answer:

the answer is part A

Step-by-step explanation:

${x}^{ \frac{12}{7} } - {x}^{ \frac{5}{7} }$

${x}^{ \frac{12 - 5}{7} }$

${x}^{ \frac{7}{7} }$

$x$

Answer 2

Complete Correct Question :-

Which of the following is equal to x?

$\rm :\longmapsto\:(a) \: {\bigg(x \bigg) }^{\dfrac{12}{7} } - {\bigg(x\bigg) }^{\dfrac{5}{7} }$

$\rm :\longmapsto\:(b) \: \sqrt[12]{ {\bigg( {x}^{4} \bigg) }^{\dfrac{1}{3} } }$

$\rm :\longmapsto\:(c) \: {\bigg( \sqrt{ {x}^{3} } \bigg) }^{\dfrac{2}{3} }$

$\rm :\longmapsto\:(d) \: {\bigg(x\bigg) }^{\dfrac{12}{7} } \times {\bigg(x\bigg) }^{\dfrac{7}{12} }$

$\large\underline{\sf{Solution-}}$

$\rm :\longmapsto\:(a) \: {\bigg(x \bigg) }^{\dfrac{12}{7} } - {\bigg(x\bigg) }^{\dfrac{5}{7} }$

we cannot simplify this further, so its not equal to x.

Consider,

$\rm :\longmapsto\:(b) \: \sqrt[12]{ {\bigg( {x}^{4} \bigg) }^{\dfrac{1}{3} } }$

$\rm = \: \sqrt[12]{ {\bigg( {x} \bigg) }^{\dfrac{4}{3} } }$

${\bigg \{ \because \: \sf \: {( {x}^{m}) }^{n} = {x}^{mn} \bigg \}}$

$\rm = \: {\bigg( {x} \bigg) }^{\dfrac{4}{3} \times \dfrac{1}{12} }$

${\bigg \{ \because \: \sf \: \sqrt[x]{y} = {\bigg(y\bigg) }^{\dfrac{1}{x} } \bigg \}}$

$\rm = \: {\bigg( {x} \bigg) }^{\dfrac{4}{9}}$

$\rm \: \ne \: x$

Consider,

$\rm :\longmapsto\:(c) \: {\bigg( \sqrt{ {x}^{3} } \bigg) }^{\dfrac{2}{3} }$

$\rm \: = \: {\bigg( {\bigg( {x}^{3} \bigg) }^{\dfrac{1}{2} } \bigg) }^{\dfrac{2}{3} }$

${\bigg \{ \because \: \sf \: \sqrt[x]{y} = {\bigg(y\bigg) }^{\dfrac{1}{x} } \bigg \}}$

$\rm \:  =  {\bigg(x\bigg) }^{\dfrac{1}{2} \times \dfrac{3}{1} \times \dfrac{2}{3} }$

$\rm \:  =  x$

So,

$\bf :\longmapsto\: \: {\bigg( \sqrt{ {x}^{3} } \bigg) }^{\dfrac{2}{3} } = x$

So,

Option (c) is correct.

Consider,

$\rm :\longmapsto\:(d) \: {\bigg(x\bigg) }^{\dfrac{12}{7} } \times {\bigg(x\bigg) }^{\dfrac{7}{12} }$

$\rm \: = {\bigg(x\bigg) }^{\dfrac{12}{7} + \dfrac{7}{12} }$

$\rm \: = {\bigg(x\bigg) }^{\dfrac{144 + 49}{84} }$

$\rm \: = {\bigg(x\bigg) }^{\dfrac{193}{84} }$

$\rm \:  \ne \: x$

Thus,

Option (c) is correct.