Question

If $m = 3$ and $n = 2$. Find the value of ${m^n} - {n^m}$.

Answer 1

Hint:
We will solve it by directly replacing the value in the given expression ${m^n} - {n^m}$ , or we can use any identity to factorize the above given expression into a simpler form then we will replace the variables to get the answer.

Complete step by step solution:
So according to the question the variables are given as $m = 3$ and $n = 2$ , now we will try to find an adequate identity to simplify or let say factorize the given expression ${m^n} - {n^m}$ , some of the adequate identities are
  ${a^n} - {b^n} = (a - b)\left( {\sum\limits_{i = 0}^{n - 1} {{a^i}{b^{n - 1 - i}}} } \right)$
 ${a^n} + {b^n} = (a + b)\left( {\sum\limits_{i = 0}^{n - 1} {{a^i}{{( - b)}^{n - 1 - i}}} } \right)$
We can see that the expression ${m^n} - {n^m}$ is quite difficult to factorize and also it doesn’t match with any other well-known identities so as the values of m,n are not too big so we will replace these values in our original expression.
So, replacing $m = 3$ and $n = 2$ in the expression ${m^n} - {n^m}$ we get
 $ = {3^2} - {2^3}$
On simplification we get,
=9-8
On subtraction we get,
=1

So, the value of the expression is 1.

Note:
Generally, we don’t directly force the replacement of values of the variables when dealing with such kinds of questions, what we do in such a situation is that we try to find an adequate identity that fits the expression given in such a question and try to simplify it to get our result.