To simplify the given expression [tex]\(\left(y^{\frac{3}{2}} x^{-\frac{1}{2}}\right)^4\)[/tex], we'll follow these steps:
1. Apply the power of a power property: The property [tex]\((a^m)^n = a^{mn}\)[/tex] allows us to multiply the exponents when raising a power to another power.
[tex]\[
\left(y^{\frac{3}{2}} x^{-\frac{1}{2}}\right)^4 = \left(y^{\frac{3}{2}}\right)^4 \cdot \left(x^{-\frac{1}{2}}\right)^4
\][/tex]
2. Multiply the exponents: For each base [tex]\(y\)[/tex] and [tex]\(x\)[/tex], we'll multiply the exponents by 4:
[tex]\[
y^{\frac{3}{2} \cdot 4} \cdot x^{-\frac{1}{2} \cdot 4}
\][/tex]
3. Simplify the exponents:
- For [tex]\(y\)[/tex]:
[tex]\[
y^{\frac{3}{2} \cdot 4} = y^{6}
\][/tex]
- For [tex]\(x\)[/tex]:
[tex]\[
x^{-\frac{1}{2} \cdot 4} = x^{-2}
\][/tex]
Thus, the simplified expression is:
[tex]\[
y^6 \cdot x^{-2}
\][/tex]
So, the correct answer is:
[tex]\[
y^6 \cdot x^{-2}
\][/tex]
