Question

Which of the following is equal to $x$?

• A. $\sqrt[12]{(x^4{)}^{\frac{1}{3}}}$
• B. $(\sqrt{x^3}{)}^{\frac{2}{3}}$
• C. $x^{\frac{12}{7}}-x^{\frac{5}{7}}$
• D. $x^{\frac{12}{7}}\times x^{\frac{7}{12}}$

Answer 1

The correct option is B

$(\sqrt{x^3}{)}^{\frac{2}{3}}$

Determine the points that's value is equal to $x$.

For point A)

$$\begin{gathered} \sqrt[12]{(x^4{)}^{\frac{1}{3}}}=\sqrt[12]{x^{4\times \frac{1}{3}}}\because {\left(a^m\right)}^x=a^{m\times x} \\ \Rightarrow =\sqrt[12]{x^{4\times \frac{1}{3}}} \\ \Rightarrow =\sqrt[12]{x^{\frac{4}{3}}} \\ \Rightarrow ={\left(x^{\frac{4}{3}}\right)}^{\frac{1}{12}} \\ \Rightarrow =x^{\frac{4}{3}\times \frac{1}{12}} \\ \Rightarrow =x^{\frac{1}{9}}\because a^{\frac{x}{n}}=\sqrt[n]{a^x} \\ \Rightarrow =\sqrt[9]{x} \end{gathered}$$

here, $\sqrt[12]{(x^4{)}^{\frac{1}{3}}}$ value is equal to $\sqrt[9]{x}$ that's not equal to $x$.

For point B)

$$\begin{gathered} {\left(\sqrt{x^3}\right)}^{\frac{2}{3}}={\left({\left(x^3\right)}^{\frac{1}{2}}\right)}^{\frac{2}{3}}\because \left(\sqrt{x}\right)=x^{\frac{1}{2}}and{\left(a^m\right)}^x=a^{m\times x} \\ \Rightarrow ={\left(x^3\right)}^{\frac{1}{2}\times \frac{2}{3}} \\ \Rightarrow ={\left(x^3\right)}^{\frac{1}{3}} \\ \Rightarrow =x^{3\times \frac{1}{3}} \\ \Rightarrow =x^1 \\ \Rightarrow =x \end{gathered}$$

here, $(\sqrt{x^3}{)}^{\frac{2}{3}}$ value is equal to $x$.

For point C)

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

here, $x^{\frac{12}{7}}-x^{\frac{5}{7}}$ is not equal to $x$.

For point D)

$$\begin{gathered} x^{\frac{12}{7}}\times x^{\frac{7}{12}}=x^{\frac{12}{7}+\frac{7}{12}}\because a^b\times a^c=a^{b+c} \\ \Rightarrow =x^{\frac{193}{84}} \end{gathered}$$

here, $x^{\frac{12}{7}}\times x^{\frac{7}{12}}$ is not equal to $x$.

Hence, the correct choice is “Option B”.