1. The problem asks to simplify the expression $\left(9^{\frac{1}{4}}\right)^{\frac{7}{2}}$ using the power of a power property. 2. The power of a power property states that $\left(a^m\right)^n = a^{m \times n}$. 3. Applying this property, we multiply the exponents: $$\left(9^{\frac{1}{4}}\right)^{\frac{7}{2}} = 9^{\frac{1}{4} \times \frac{7}{2}}$$ 4. Multiply the exponents: $$\frac{1}{4} \times \frac{7}{2} = \frac{7}{8}$$ 5. So the expression simplifies to: $$9^{\frac{7}{8}}$$ 6. Since $9 = 3^2$, rewrite the base: $$9^{\frac{7}{8}} = \left(3^2\right)^{\frac{7}{8}}$$ 7. Apply the power of a power property again: $$\left(3^2\right)^{\frac{7}{8}} = 3^{2 \times \frac{7}{8}} = 3^{\frac{14}{8}}$$ 8. Simplify the exponent fraction: $$\frac{14}{8} = \frac{7}{4}$$ 9. Final simplified expression: $$3^{\frac{7}{4}}$$ Therefore, the simplified form of $\left(9^{\frac{1}{4}}\right)^{\frac{7}{2}}$ is $3^{\frac{7}{4}}$.