Certainly! Let's simplify the given expression step-by-step using the properties of exponents:
Given expression:
[tex]\[
\left( y^{\frac{3}{2}} x^{-\frac{1}{2}} \right)^4
\][/tex]
Using the properties of exponents, specifically the power of a product property [tex]\((a^m \cdot b^n)^p = a^{m \cdot p} \cdot b^{n \cdot p}\)[/tex], we can distribute the exponent 4 to each term inside the parentheses.
First, apply the exponent to the [tex]\(y\)[/tex] term:
[tex]\[
\left( y^{\frac{3}{2}} \right)^4 = y^{\frac{3}{2} \cdot 4}
\][/tex]
Next, apply the exponent to the [tex]\(x\)[/tex] term:
[tex]\[
\left( x^{-\frac{1}{2}} \right)^4 = x^{-\frac{1}{2} \cdot 4}
\][/tex]
Now, let's calculate the exponents:
For [tex]\(y\)[/tex]:
[tex]\[
\frac{3}{2} \cdot 4 = 6
\][/tex]
So,
[tex]\[
y^{\frac{3}{2} \cdot 4} = y^6
\][/tex]
For [tex]\(x\)[/tex]:
[tex]\[
-\frac{1}{2} \cdot 4 = -2
\][/tex]
So,
[tex]\[
x^{-\frac{1}{2} \cdot 4} = x^{-2}
\][/tex]
Combining these results, the simplified expression is:
[tex]\[
y^6 \cdot x^{-2}
\][/tex]
Therefore, the simplified form of the expression [tex]\(\left( y^{\frac{3}{2}} x^{-\frac{1}{2}} \right)^4\)[/tex] is:
[tex]\[
y^6 \cdot x^{-2}
\][/tex]
