https://r2.erweima.ai/mathbot/image/1725934705275b5tpwqt4.jpg

The expression shown in the image is:

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

Let's simplify this step by step.

### Step 1: Simplify the numerator
The numerator is \(\left( 2x^{-3}y^2 \right)^3\). We apply the power of a product rule: \((ab)^n = a^n \cdot b^n\).

- \(2^3 = 8\)
- \(\left(x^{-3}\right)^3 = x^{-9}\)
- \(\left(y^2\right)^3 = y^6\)

Thus, the numerator becomes:
\[
8x^{-9}y^6
\]

### Step 2: Simplify the denominator
The denominator is \(2x^3y^{-3}\).

### Step 3: Combine the simplified numerator and denominator
Now, express the fraction as:
\[
\frac{8x^{-9}y^6}{2x^3y^{-3}}
\]

Simplify by dividing the coefficients (8 divided by 2 is 4) and applying the laws of exponents for the variables:

- For \(x\): \(\frac{x^{-9}}{x^3} = x^{-9 - 3} = x^{-12}\)
- For \(y\): \(\frac{y^6}{y^{-3}} = y^{6 - (-3)} = y^{9}\)

Thus, the final simplified expression is:
\[
4x^{-12}y^9
\]

Or, equivalently:
\[
\frac{4y^9}{x^{12}}
\]

Would you like further details or have any questions?

### Related Questions:
1. How do you apply the power of a product rule in algebraic expressions?
2. What are the rules for dividing exponents with the same base?
3. How does the power rule for exponents affect negative exponents?
4. Can you simplify expressions with multiple variables in the same manner?
5. How can you convert a negative exponent into a positive one in a fraction?

### Tip:
When simplifying expressions with exponents, always remember to handle coefficients separately from variables and apply the exponent rules step by step.

Simplify the expression (2x^(-3) y^2)^3 / (2x^3 y^(-3))

Learn how to simplify the algebraic expression (2x^(-3) y^2)^3 / (2x^3 y^(-3)) with a step-by-step solution. This problem covers key exponent rules and algebraic simplification, suitable for students in Grades 9-11.

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