MCQ
If $a, m, n$ are positive ingegers, then $\{\sqrt[m]{\sqrt[n]{a}}\}^{m n}$ is equal to
- A$a ^{ nm }$
- ✓$a$
- C$a^{\frac{m}{n}}$
- D$1$
✓
Answer
Correct option: B.
$a$
Find the value of $\{\sqrt[m]{\sqrt[n]{a}}\}^{ mn }$
So,
$\{\sqrt[m]{\sqrt[n]{a}}\}^{m n}=\left\{\sqrt[m]{a^{\frac{1}{n}}}\right\}^{m n}$
$=\left\{a^{\frac{1}{n} \times \frac{1}{m}}\right\}^{m n}$
$=\left\{a^{\frac{1}{n} \times \frac{1}{m}} \times m \times n\right\}$
$\Rightarrow\{\sqrt[m]{\sqrt[n]{a}}\}=\left\{a^{\frac{1}{n} \times \frac{1}{m}} \times m \times n\right\}$
$\Rightarrow\{\sqrt[m]{\sqrt[n]{a}}\}=a$
Hence the correct choice is $b$.
So,
$\{\sqrt[m]{\sqrt[n]{a}}\}^{m n}=\left\{\sqrt[m]{a^{\frac{1}{n}}}\right\}^{m n}$
$=\left\{a^{\frac{1}{n} \times \frac{1}{m}}\right\}^{m n}$
$=\left\{a^{\frac{1}{n} \times \frac{1}{m}} \times m \times n\right\}$
$\Rightarrow\{\sqrt[m]{\sqrt[n]{a}}\}=\left\{a^{\frac{1}{n} \times \frac{1}{m}} \times m \times n\right\}$
$\Rightarrow\{\sqrt[m]{\sqrt[n]{a}}\}=a$
Hence the correct choice is $b$.
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