Question

The value of ${\left[\sqrt[3]{\sqrt[6]{{\mathrm{a}}^9}}\right]}^4{\left[\sqrt[6]{\sqrt[3]{{\mathrm{a}}^9}}\right]}^4$ is :

• A. ${\mathrm{a}}^{16}$
• B. ${\mathrm{a}}^{12}$
• C. ${\mathrm{a}}^8$
• D. ${\mathrm{a}}^4$

Answer 1

The correct option is D

${\mathrm{a}}^4$

Solution:

Simplify the given expression:

$$\begin{gathered} {\left[\sqrt[3]{\sqrt[6]{{\mathrm{a}}^9}}\right]}^4{\left[\sqrt[6]{\sqrt[3]{{\mathrm{a}}^9}}\right]}^4 \\ ={\left[\sqrt[3]{{\left({\left(a\right)}^9\right)}^{\frac{1}{6}}}\right]}^4{\left[\sqrt[6]{{\left({\left(\mathrm{a}\right)}^9\right)}^{\frac{1}{3}}}\right]}^4\mathbf{}\mathbf{[}\mathbf{\because }\sqrt[\mathbf{q}]{\mathbf{p}}\mathbf{=}{\mathbf{\left(}\mathbf{p}\mathbf{\right)}}^{\frac{\mathbf{1}}{\mathbf{q}}}\mathbf{]} \\ ={\left[{\left({\left({\left(a\right)}^9\right)}^{\frac{1}{6}}\right)}^{\frac{1}{3}}\right]}^4{\left[\left[{\left({\left({\left(a\right)}^9\right)}^{\frac{1}{3}}\right)}^{\frac{1}{6}}\right]\right]}^4 \\ ={\left(a^{9\times \frac{1}{6}\times \frac{1}{3}}\right)}^4{\left(a^{9\times \frac{1}{3}\times \frac{1}{6}}\right)}^4\mathbf{[}\mathbf{\because }{\mathbf{\left(}{\mathbf{a}}^{\mathbf{n}}\mathbf{\right)}}^{\mathbf{m}}\mathbf{=}{\mathbf{a}}^{\mathbf{m}\mathbf{n}}\mathbf{]} \\ ={\left(a^{\frac{9}{18}}\right)}^4{\left(a^{\frac{9}{18}}\right)}^4 \\ ={\left(a^{\frac{1}{2}}\right)}^4{\left(a^{\frac{1}{2}}\right)}^4 \\ =\left(a^{\frac{4}{2}}\right)\left(a^{\frac{4}{2}}\right)\mathbf{} \\ =\left(a^2\right)\left(a^2\right) \\ ={\mathit{a}}^{\mathbf{4}}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{\left(}{\mathbf{v}}^{\mathbf{z}}\mathbf{\right)}\mathbf{\left(}{\mathbf{v}}^{\mathbf{z}}\mathbf{\right)}\mathbf{=}{\mathbf{v}}^{\mathbf{2}\mathbf{z}}\mathbf{]} \end{gathered}$$

Final answer: Hence, option D is the correct answer.