The given mathematical expression is:

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

We need to simplify this step by step.

Step 1: Simplify the fraction inside the parentheses

The expression inside the parentheses can be written as:

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

This becomes:

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

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

Now apply the exponent $-\frac{3}{4}$ to the simplified expression:

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

This will affect both the numerator and the denominator:

$2^{-\frac{3}{4}} \times (x^5)^{-\frac{3}{4}} \times (y^{-2/3})^{-\frac{3}{4}}$

Simplifying each term:

$2^{-\frac{3}{4}} = \frac{1}{2^{3/4}}, \quad (x^5)^{-\frac{3}{4}} = x^{-15/4}, \quad (y^{-2/3})^{-\frac{3}{4}} = y^{\frac{1}{2}}$

Thus, the expression becomes:

$\frac{1}{2^{3/4}} \times x^{-15/4} \times y^{1/2}$

Step 3: Rewrite the expression in a more familiar form

This can be written as:

$\frac{y^{1/2}}{2^{3/4} x^{15/4}}$

Now we convert this back to the square root form:

$\frac{\sqrt{y}}{\sqrt[4]{2^3 x^{15}}}$

This simplifies to:

$\frac{\sqrt{y}}{\sqrt[4]{8 x^{15}}}$

Step 4: Compare with the options

From the given options, the closest match is B:

$x \sqrt{\frac{y}{x}}$

Thus, the correct answer is B.

Would you like more details on any of the steps? Here are five related questions for further practice:

1. How do you simplify expressions with fractional exponents?
2. How do you handle negative exponents when simplifying?
3. What are the rules for multiplying and dividing exponents?
4. How do you convert between radical and exponent forms?
5. How do you simplify nested fractions involving variables?

Tip: When working with exponents, always simplify the powers of variables first before applying any further operations.