Select the exponent(s) which is(are) equal to $(a^{m})^{n} .$

a^{m n}

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(a^{n})^{m}

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a^{m^{n}}

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a^{n^{m}}

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Solution

The correct option is B $(a^{n})^{m}$
$(a^{m})^{n}$ is a question of repeated exponentiation.

Let's take an example first.
$(2^{3})^{4}$ represents $4$ times multiplication of $2^{3},$ i.e.,
$(2^{3})^{4}$
$= 2^{3} \times 2^{3} \times 2^{3} \times 2^{3}$

$= 2^{3 + 3 + 3 + 3}$
$(∵ a^{m} \times a^{n} \times a^{p} \times a^{q} = a^{m + n + p + q})$

$= 2^{12}$
$\Rightarrow (2^{3})^{4} = 2^{3 \times 4} - ----------- -$

By generalizing the above finding, we get:
$(a^{m})^{n} = a^{m \times n} = a^{m n}$
Similarly, $(a^{n})^{m} = a^{n \times m} = a^{n m} = a^{m n}$

Therefore, $a^{m n}$ and $(a^{n})^{m}$ are equal to $(a^{m})^{n}$ .

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