To simplify the given expression
[tex]\[
\left(\frac{x^{\frac{3}{4}}}{y^{-\frac{1}{3}}}\right)^6,
\][/tex]
follow these steps:
### Step 1: Understanding the Expression
The expression inside the parenthesis is
[tex]\[
\frac{x^{\frac{3}{4}}}{y^{-\frac{1}{3}}}.
\][/tex]
### Step 2: Simplifying the Denominator
Recall that a negative exponent indicates a reciprocal, so [tex]\( y^{-\frac{1}{3}} \)[/tex] can be rewritten as [tex]\( \frac{1}{y^{\frac{1}{3}}} \)[/tex]. This gives:
[tex]\[
\frac{x^{\frac{3}{4}}}{\frac{1}{y^{\frac{1}{3}}}} = x^{\frac{3}{4}} \cdot y^{\frac{1}{3}}.
\][/tex]
### Step 3: Applying the Exponentiation
Next, apply the 6th power to each term inside the parenthesis:
[tex]\[
\left(x^{\frac{3}{4}} \cdot y^{\frac{1}{3}}\right)^6.
\][/tex]
Using the rule [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex], we can split this into:
[tex]\[
\left(x^{\frac{3}{4}}\right)^6 \cdot \left(y^{\frac{1}{3}}\right)^6.
\][/tex]
### Step 4: Power of a Power
Use the rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] to simplify:
[tex]\[
\left(x^{\frac{3}{4}}\right)^6 = x^{\frac{3}{4} \cdot 6} = x^{\frac{18}{4}} = x^{4.5},
\][/tex]
and
[tex]\[
\left(y^{\frac{1}{3}}\right)^6 = y^{\frac{1}{3} \cdot 6} = y^2.
\][/tex]
### Step 5: Combining the Results
Putting these simplified parts together, we get:
[tex]\[
x^{4.5} \cdot y^2.
\][/tex]
### Conclusion
Thus, the simplified form of the given expression is:
[tex]\[
x^{4.5} y^2.
\][/tex]
Among the given options, this corresponds to:
[tex]\[
x^{\frac{9}{2}} y^2.
\][/tex]
