https://file.aiquickdraw.com/mathbot/file/1725486122682dsdl74if.jpg

The expression in the image is:

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

To simplify this expression, let's follow the steps below:

1. **Apply the Negative Exponent Rule**: Recall that \( a^{-n} = \frac{1}{a^n} \). This rule applies to both the fraction and the exponent:
   
   \[
   \left(\frac{x^{-3}}{y^3}\right)^{-7} = \left(\frac{y^3}{x^{-3}}\right)^{7}
   \]

2. **Simplify the Fraction Inside the Parentheses**: Since \( x^{-3} = \frac{1}{x^3} \), the fraction becomes:

   \[
   \frac{y^3}{\frac{1}{x^3}} = y^3 \cdot x^3 = x^3y^3
   \]

3. **Apply the Exponent to Each Term**: Distribute the exponent of 7 to both \( x^3 \) and \( y^3 \):

   \[
   \left(x^3y^3\right)^7 = (x^3)^7 \cdot (y^3)^7 = x^{21} \cdot y^{21}
   \]

Thus, the simplified form of the expression is:

\[
x^{21}y^{21}
\]

Would you like any more details or have any questions?

Here are five related questions:
1. How would you simplify \( \left(\frac{x^2}{y^{-5}}\right)^{-3} \)?
2. What happens when you have a negative exponent outside the parentheses for a fraction?
3. How do you simplify expressions with multiple negative exponents?
4. What is the general rule for distributing exponents over products?
5. Can you simplify \( \left(\frac{x^m}{y^n}\right)^{-k} \) for general \( m \), \( n \), and \( k \)?

**Tip:** Always remember to handle negative exponents carefully by applying the rule \( a^{-n} = \frac{1}{a^n} \) before simplifying the expression further.

Simplify the expression (x^(-3) / y^3)^(-7).

Explore the steps to simplify the expression (x^(-3) / y^3)^(-7) using the negative exponent rule and power of a power rule. Perfect for students in Grades 9-10 learning advanced algebraic concepts.

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