1. The problem is to verify the expression $\left(\frac{9}{2}\right)^2 = 81$. 2. Start by squaring the fraction $\frac{9}{2}$: $$\left(\frac{9}{2}\right)^2 = \frac{9^2}{2^2} = \frac{81}{4}.$$ 3. The result of squaring $\frac{9}{2}$ is $\frac{81}{4}$, not 81. 4. Therefore, $\left(\frac{9}{2}\right)^2 \neq 81$; it equals $\frac{81}{4}$. 5. If you want the expression to equal 81, the base must be $\sqrt{81} = 9$, not $\frac{9}{2}$. Final answer: $$\left(\frac{9}{2}\right)^2 = \frac{81}{4} \neq 81.$$