https://file.aiquickdraw.com/mathbot/file/1725417433171zosun2t2.jpg

The expression shown in the image is:

\[
\left(\frac{9z}{8z^6}\right)^3
\]

Let's simplify and solve this step by step.

### Step 1: Simplify the fraction inside the parentheses
The expression inside the parentheses is:

\[
\frac{9z}{8z^6}
\]

We can simplify this by dividing the terms with the same base (which is \( z \)):

\[
\frac{9}{8} \cdot \frac{z}{z^6} = \frac{9}{8} \cdot z^{1-6} = \frac{9}{8} \cdot z^{-5} = \frac{9}{8z^5}
\]

So the expression simplifies to:

\[
\left(\frac{9}{8z^5}\right)^3
\]

### Step 2: Apply the exponent to the fraction
Now, we need to raise the simplified expression to the power of 3:

\[
\left(\frac{9}{8z^5}\right)^3 = \frac{9^3}{(8z^5)^3}
\]

### Step 3: Calculate the powers
Compute the powers of the constants and the variable:

\[
9^3 = 729 \quad \text{and} \quad (8z^5)^3 = 8^3 \cdot z^{15} = 512 \cdot z^{15}
\]

So, the final expression is:

\[
\frac{729}{512z^{15}}
\]

### Final Answer

The simplified expression is:

\[
\boxed{\frac{729}{512z^{15}}}
\]

Do you have any questions, or would you like more details?

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Here are five related questions that expand on this problem:

1. How would the solution change if the exponent were different, say 4 instead of 3?
2. What if the variable \(z\) had an initial negative exponent, like \(z^{-2}\) instead of \(z\)?
3. How does this problem change if you were to simplify \(\left(\frac{az^m}{bz^n}\right)^p\) in general terms?
4. What would be the effect of adding or subtracting terms in the numerator or denominator?
5. How would the solution differ if there were additional variables, such as \(y\) or \(x\), in the expression?

**Tip:** Always simplify expressions inside parentheses before applying exponents; it makes calculations easier and reduces the chances of errors.

Simplify the expression \(\left(\frac{9z}{8z^6}\right)^3\) and provide the final simplified form.

Learn how to simplify the algebraic expression \(\left(\frac{9z}{8z^6}\right)^3\). This problem involves key mathematical concepts such as exponentiation, simplifying fractions, and algebraic manipulation. Suitable for students in Grades 8-10.

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