The question in the uploaded image asks you to simplify the given expression:

$x^{-\frac{3}{4}} \cdot x^{\frac{9}{4}}$

Solution:

To simplify this, we apply the product of powers rule: $a^m \cdot a^n = a^{m+n}$

Here, the bases are the same ($x$), so we add the exponents:

$-\frac{3}{4} + \frac{9}{4} = \frac{-3 + 9}{4} = \frac{6}{4} = \frac{3}{2}$

Thus, the simplified expression is: $x^{\frac{3}{2}}$

Final Answer:

The correct option is C: $x^{\frac{3}{2}}$.

Would you like further clarification? Here are 5 related questions for further practice:

1. Simplify: $x^{\frac{1}{3}} \cdot x^{\frac{5}{3}}$.
2. Simplify: $(y^2)^{\frac{3}{4}}$.
3. Simplify: $a^{-\frac{1}{2}} \cdot a^{\frac{7}{2}}$.
4. Solve for $x$ if $x^{\frac{3}{2}} = 8$.
5. Simplify: $x^{\frac{4}{3}} \cdot x^{-\frac{7}{3}}$.

Tip:

Always add or subtract exponents when multiplying or dividing terms with the same base, respectively.