Question

Simplify the expression. The simplified expression should have no negative exponents. $$ \left(\frac{a^{9}}{a^{5}}\right)^{-1} $$

Short answer

The simplified expression of \(\left(\frac{a^{9}}{a^{5}}\right)^{-1}\) is \( \frac{1}{a^4}\) .

Step-by-step solution

Understand the expression

The provided expression is \(\left(\frac{a^{9}}{a^{5}}\right)^{-1}\). Applying the rule -- when dividing numbers with the same base, subtract their exponents-- helps to simplify the expression within the parentheses.

Simplify the base

Apply the law of exponents and subtract 5 from 9 in the exponent, which gives \(a^{9-5} = a^4\). So, the expression becomes \( (a^{4})^{-1} \) .

Address the negative exponent

A negative exponent simply means we find the reciprocal of the base. Hence, \( (a^{4})^{-1} = \frac{1}{a^4} \).