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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!

Hasil dari (3y/4)^-2 * (3zx^2y^-1z / 2^2x^-1y^3z^-2)^2 adalah...

Learn how to simplify the expression involving negative powers and fractions: (3y/4)^-2 * (3zx^2y^-1z / 2^2x^-1y^3z^-2)^2. This example covers exponents, fraction simplification, and algebraic manipulation. Suitable for high school students in Grades 9-12.

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