The image contains a mathematical expression and multiple-choice answers. The expression is:

$\left( \frac{\frac{2}{x^3y^{\frac{4}{3}}}}{x^2 y^3} \right)^{-\frac{3}{4}}$

To simplify this expression, let's break it down step by step:

Step 1: Simplify the expression inside the brackets

The expression inside the brackets is a fraction:

$\frac{\frac{2}{x^3 y^{\frac{4}{3}}}}{x^2 y^3}$

This can be rewritten as:

$\frac{2}{x^3 y^{\frac{4}{3}}} \times \frac{1}{x^2 y^3} = \frac{2}{x^3 y^{\frac{4}{3}}} \times \frac{1}{x^2 y^3}$

Multiplying the $x$ and $y$ terms together:

$\frac{2}{x^{3+2} y^{\frac{4}{3} + 3}} = \frac{2}{x^5 y^{\frac{13}{3}}}$

Step 2: Apply the outer exponent $-\frac{3}{4}$

Next, apply the exponent $-\frac{3}{4}$ to both the numerator and denominator:

$\left( \frac{2}{x^5 y^{\frac{13}{3}}} \right)^{-\frac{3}{4}} = 2^{-\frac{3}{4}} x^{5 \times \frac{3}{4}} y^{\frac{13}{3} \times \frac{3}{4}}$

Simplifying the powers:

$= 2^{-\frac{3}{4}} x^{\frac{15}{4}} y^{\frac{-39}{12}} = 2^{-\frac{3}{4}} x^{\frac{15}{4}} y^{-\frac{13}{4}}$

Step 3: Simplify the constants and terms

Now, rewrite the final expression as:

$\frac{x^{\frac{15}{4}}}{2^{\frac{3}{4}} y^{\frac{13}{4}}}$

Step 4: Match with answer choices

The expression is closest to the form of choice C:

$\sqrt{x^2 y}$

Thus, the correct answer is C.

Would you like further details on the steps or more related problems? Here are five questions that relate:

1. How can you simplify a complex fraction like the one in this problem?
2. What properties of exponents are useful when simplifying expressions like this?
3. How does applying a fractional exponent work when both the numerator and denominator have variables?
4. What is the general process for simplifying powers of variables?
5. How can we estimate or understand fractional exponents with negative values?

Tip: Always simplify inside the brackets first before applying outer exponents.