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The given expression is:

\[
\left( \frac{2x^{-3} y^7}{z^{-1}} \right)^3
\]

### Step-by-Step Solution:

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

\[
\frac{2x^{-3} y^7}{z^{-1}}
\]

Since \(z^{-1} = \frac{1}{z}\), dividing by \(z^{-1}\) is equivalent to multiplying by \(z\). Hence:

\[
\frac{2x^{-3} y^7}{z^{-1}} = 2x^{-3} y^7 \cdot z^1 = 2x^{-3} y^7 z
\]

#### Step 2: Apply the cube (power of 3) to each term.
Using the power rule \((ab)^3 = a^3 b^3\), distribute the exponent of 3 to all parts:

\[
\left(2x^{-3} y^7 z\right)^3 = 2^3 \cdot (x^{-3})^3 \cdot (y^7)^3 \cdot (z^1)^3
\]

#### Step 3: Simplify each term.
- \(2^3 = 8\)
- \((x^{-3})^3 = x^{-9}\)
- \((y^7)^3 = y^{21}\)
- \((z^1)^3 = z^3\)

So the expression becomes:

\[
8 \cdot x^{-9} \cdot y^{21} \cdot z^3
\]

#### Step 4: Write the final simplified result.
The simplified expression is:

\[
8x^{-9} y^{21} z^3
\]

If you prefer positive exponents, you can rewrite \(x^{-9}\) as \(\frac{1}{x^9}\), but usually both forms are acceptable:

\[
8 \cdot \frac{y^{21} z^3}{x^9}
\]

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

Here are some related questions to expand your understanding:

1. What are the rules for multiplying powers with the same base?
2. How do negative exponents affect the placement of variables in fractions?
3. What is the general rule for distributing exponents over fractions?
4. How would the result change if one or more terms had a different negative exponent?
5. Can you simplify an expression like \(\left(\frac{x^{-1}y}{z^2}\right)^4\) on your own?

**Tip:** Always simplify exponents step-by-step to avoid missing small changes, especially when dealing with negative values!

Simplify the expression (2x^-3 * y^7 / z^-1)^3.

Learn how to simplify the algebraic expression (2x^-3 * y^7 / z^-1)^3 using exponent rules. This step-by-step guide covers key algebraic concepts, including the laws of exponents and the division rule. Suitable for students in Grades 8-10.

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