The given expression appears to be a combination of two parts involving powers and fractions:

$\left(\frac{3y}{4}\right)^{-2} \left( \frac{3z x^2 y^{-1} z}{2^2 x^{-1} y^3 z^{-2}} \right)^2$

Let's simplify the expression step by step:

Step 1: Simplify $\left(\frac{3y}{4}\right)^{-2}$

Applying the power rule:

$\left(\frac{3y}{4}\right)^{-2} = \left(\frac{4}{3y}\right)^2 = \frac{16}{9y^2}$

Step 2: Simplify the second fraction

The second fraction inside the parentheses is:

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

Now, apply the laws of exponents to simplify:

• $z^2 / z^{-2} = z^{2 - (-2)} = z^4$
• $x^2 / x^{-1} = x^{2 - (-1)} = x^3$
• $y^{-1} / y^3 = y^{-1 - 3} = y^{-4}$

This gives:

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

Step 3: Apply the square

Now, square the simplified fraction:

$\left( \frac{3z^4 x^3 y^{-4}}{4} \right)^2 = \frac{9z^8 x^6 y^{-8}}{16}$

Step 4: Combine both parts

Now multiply the two parts:

$\frac{16}{9y^2} \times \frac{9z^8 x^6 y^{-8}}{16} = \frac{9z^8 x^6 y^{-8}}{9y^2} = \frac{z^8 x^6}{y^{10}}$

Final answer:

The correct option is d: $\frac{9x^6 z^8}{y^{10}}$

Do you want any further details on this solution or explanation of the steps? Here are five related questions:

1. How do exponent rules apply when dividing two powers of the same base?
2. What happens when we raise a fraction to a negative power?
3. How do you simplify complex fractions involving exponents?
4. What are the steps to apply when multiplying two terms with exponents?
5. How does squaring a fraction affect both the numerator and denominator?

Tip: Always remember to apply exponent rules carefully when dealing with negative and fractional exponents to avoid errors!