To simplify the expression [tex]\(\left(\sqrt[4]{a^5 b^3}\right)^2\)[/tex], we will follow these steps:
1. Simplify Inside the Radical:
[tex]\[
\sqrt[4]{a^5 b^3}
\][/tex]
This can be rewritten using fractional exponents:
[tex]\[
(a^5 b^3)^{1/4}
\][/tex]
2. Distribute the Exponent:
Using the property [tex]\((xy)^n = x^n y^n\)[/tex], we distribute the [tex]\(\frac{1}{4}\)[/tex] exponent:
[tex]\[
\left(a^5\right)^{1/4} \left(b^3\right)^{1/4}
\][/tex]
Simplify the individual terms:
[tex]\[
a^{5/4} \cdot b^{3/4}
\][/tex]
3. Raise the Entire Expression to the Power of 2:
Now take the expression [tex]\(a^{5/4} \cdot b^{3/4}\)[/tex] and raise it to the power of 2:
[tex]\[
\left(a^{5/4} b^{3/4}\right)^2
\][/tex]
4. Distribute the Power:
Using the property [tex]\((x^m)^n = x^{mn}\)[/tex], we distribute the exponent of 2:
[tex]\[
(a^{5/4})^2 \cdot (b^{3/4})^2
\][/tex]
This results in:
[tex]\[
a^{(5/4) \cdot 2} \cdot b^{(3/4) \cdot 2}
\][/tex]
5. Simplify the Exponents:
Perform the multiplication in the exponents:
[tex]\[
a^{5/2} \cdot b^{3/2}
\][/tex]
Thus, the simplified form of the given expression is:
[tex]\[
\boxed{a^{5/2} b^{3/2}}
\][/tex]
From the provided options, the correct one is:
c. [tex]\(a^{\frac{5}{2}} b^{\frac{3}{2}}\)[/tex]
