Question

If \[{({a^m})^n} = {a^{{m^n}}}\], then express \[m\] in terms of \[n\].

Answer 1

Hint: To find the answer of this question we have to compare the exponent in given identities. We use the property: \[{a^m} \times {a^n} = {a^{m + n}}\] and \[\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}\]. Where a is base while m, and n are power or exponent of the given exponential.

Complete step by step answer:
According to the question:-
Given that, \[{({a^m})^n} = {a^{{m^n}}}\]
\[{a^{mn}} = {a^{{m^n}}}\]
The bases of the exponentials are identical. So, we can compare exponents on both sides.
\[mn = {m^n}\]………………… (i)
Dividing eqn (i) by \[m\]we get,
\[ \Rightarrow n = \dfrac{{{m^n}}}{m}\]
\[ \Rightarrow n = {m^{(n - 1)}}\]

Hence, \[m = {n^{\left( {\dfrac{1}{{n - 1}}} \right)}}\]

Note:
In order to tackle such kinds of problems of exponential One must have a basic understanding of exponential and its properties.
Exponent of a number indicates how many times the same number multiply by itself .For example: \[{5^3} = 5 \times 5 \times 5\].
Some important properties of exponentials:-
\[
  {a^0} = 1 \\
  {a^m} \times {a^n} = {a^{m + n}} \\
  \dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}} \\
\]
\[{\left( {{a^m}} \right)^n} = {a^{mn}}\]
One should note that whenever we compare the two exponentials, we should well know about the base and power of the exponent. If the bases of two exponentials are equal then power will also be equal to each other. i.e. if \[{a^k} = {a^p}\] then \[k = p\].