To simplify the expression [tex]\(\left(9^{\frac{1}{4}}\right)^{\frac{7}{2}}\)[/tex] using the power of a power property, we'll follow these steps:
1. Recall the power of a power property: [tex]\((a^m)^n = a^{m \cdot n}\)[/tex].
2. Apply the property to the expression: [tex]\(\left(9^{\frac{1}{4}}\right)^{\frac{7}{2}}\)[/tex]:
[tex]\[
\left(9^{\frac{1}{4}}\right)^{\frac{7}{2}} = 9^{\left(\frac{1}{4} \times \frac{7}{2}\right)}
\][/tex]
3. Multiply the exponents:
[tex]\[
\frac{1}{4} \times \frac{7}{2} = \frac{7}{8}
\][/tex]
4. Write the expression with the new exponent:
[tex]\[
9^{\frac{7}{8}}
\][/tex]
Therefore, the simplified numeric expression is:
[tex]\[
9^{\frac{7}{8}}
\][/tex]
Finally, evaluating [tex]\(9^{\frac{7}{8}}\)[/tex] results in approximately:
[tex]\[
6.8385211708643325
\][/tex]
So, [tex]\(\left(9^{\frac{1}{4}}\right)^{\frac{7}{2}} = 6.8385211708643325\)[/tex].
