Simplify using positive indices: (xy^-3)^-2

To simplify the expression \((xy^{-3})^{-2}\) using positive indices, follow these steps:

1. **Apply the power of a power rule**: This rule states that \((a^m)^n = a^{mn}\). Apply this rule to each term inside the parentheses:

   \[
   (xy^{-3})^{-2} = x^{-2} \cdot (y^{-3})^{-2}
   \]

2. **Simplify each term**:
   - For the \(x\) term: \(x^{-2}\) remains as is.
   - For the \(y^{-3}\) term: \((y^{-3})^{-2} = y^{(-3) \times (-2)} = y^6\).

3. **Combine the simplified terms**:
   \[
   x^{-2} \cdot y^6
   \]

4. **Convert to positive indices**: To express \(x^{-2}\) with a positive exponent, recall that \(a^{-m} = \frac{1}{a^m}\):

   \[
   x^{-2} \cdot y^6 = \frac{y^6}{x^2}
   \]

So, the simplified expression is:
\[
\frac{y^6}{x^2}
\]

Would you like more details or have any questions?

Here are five related questions for further practice:
1. Simplify the expression \((x^{-1}y^2)^{-3}\) using positive indices.
2. How would you simplify \(\frac{x^{-4}y^3}{x^{-2}y^{-5}}\) using positive indices?
3. What is the result of \((a^{-2}b^3)^2\) expressed with positive indices?
4. How would you simplify the expression \((3x^{-2})^3\) using positive indices?
5. Simplify the expression \(\left(\frac{x^3}{y^{-2}}\right)^{-1}\) using positive indices.

**Tip:** When simplifying expressions with negative exponents, remember that \(a^{-m} = \frac{1}{a^m}\). This will help you convert all exponents to positive indices easily.

Learn how to simplify the expression (xy^-3)^-2 using positive indices. Follow step-by-step instructions to convert negative exponents to positive exponents and simplify the expression to y^6 / x^2. Suitable for Grades 9-12.

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