- First, simplify the internal terms of each fraction by combining powers of \(x\) and \(y\).
- Apply the outer exponents \(-2\) and \(2\) to these simplified internal expressions.
- Multiply the resulting terms by grouping coefficients, powers of \(x\), and powers of \(y\).
- The final result after combining and simplifying the exponents is \(\boxed{\frac{4}{9} x^{-6}}\).
Explanation
- Problem Analysis and Strategy
The expression we need to simplify is \(\left(\frac{x^5 y^{-1}}{6 y^2}\right)^{-2} \cdot \left(\frac{x^2 \cdot y^{-2}}{9 y}\right)^2\). Our strategy is to simplify each part of the expression separately using exponent laws before multiplying them together to obtain the final form \(A x^m y^n\). Exponent laws we will apply include \((ab)^p = a^p b^p\), \((a^p)^q = a^{pq}\), and \(a^p \cdot a^q = a^{p+q}\).
- Simplifying the First Term
First, we simplify the first term \(\left(\frac{x^5 y^{-1}}{6 y^2}\right)^{-2}\). Inside the parentheses, we combine the \(y\) terms: \(\frac{x^5 y^{-1}}{6 y^2} = \frac{x^5}{6 y^{2-(-1)}} = \frac{x^5}{6 y^3} = \frac{1}{6} x^5 y^{-3}\). Now applying the power \(-2\) to this entire expression:
\[\left(\frac{1}{6} x^5 y^{-3}\right)^{-2} = \left(\frac{1}{6}\right)^{-2} (x^5)^{-2} (y^{-3})^{-2} = (6^2) (x^{-10}) (y^6) = 36 x^{-10} y^6.\]
Wait, let's recheck the coefficient: \(\left(\frac{x^5 y^{-1}}{6 y^2}\right)^{-2} = \frac{(x^5 y^{-1})^{-2}}{(6 y^2)^{-2}} = \frac{x^{-10} y^2}{6^{-2} y^{-4}} = 36 x^{-10} y^{2-(-4)} = 36 x^{-10} y^6\).
Actually, let's recalculate \(\frac{x^5}{6 y^3}\) raised to the power of \(-2\):
\[\left(\frac{x^5}{6 y^3}\right)^{-2} = \left(\frac{6 y^3}{x^5}\right)^2 = \frac{36 y^6}{x^{10}} = 36 x^{-10} y^6.\]
Wait, the \(\frac{1}{6}\) in my previous mental step was \(6^{-1}\).
Let's redo: \(\left(\frac{x^5 y^{-1}}{6 y^2}\right)^{-2} = \left(\frac{x^5}{6 y^3}\right)^{-2} = \left(\frac{6 y^3}{x^5}\right)^2 = \frac{36 y^6}{x^{10}} = 36 x^{-10} y^6\).
Correct.
- Simplifying the Second Term
Next, we simplify the second term \(\left(\frac{x^2 y^{-2}}{9 y}\right)^2\). Inside the parentheses, combine the \(y\) terms: \(\frac{x^2 y^{-2}}{9 y} = \frac{x^2}{9 y^{1-(-2)}} = \frac{x^2}{9 y^3} = \frac{1}{9} x^2 y^{-3}\).
Now apply the power \(2\):
\[\left(\frac{x^2}{9 y^3}\right)^2 = \frac{(x^2)^2}{9^2 (y^3)^2} = \frac{x^4}{81 y^6} = \frac{1}{81} x^4 y^{-6}.\]
This simplifies perfectly.
- Combining the Expressions
Now multiply the two simplified expressions:
\[(36 x^{-10} y^6) \cdot \left(\frac{1}{81} x^4 y^{-6}\right) = \left(36 \cdot \frac{1}{81}\right) \cdot (x^{-10} \cdot x^4) \cdot (y^6 \cdot y^{-6}).\]
Simplify the coefficients: \(\frac{36}{81} = \frac{4 \cdot 9}{9 \cdot 9} = \frac{4}{9}\).
Simplify the variables: \(x^{-10} \cdot x^4 = x^{-10+4} = x^{-6}\) and \(y^6 \cdot y^{-6} = y^{6-6} = y^0 = 1\).
The final expression is \(\frac{4}{9} x^{-6}\).
- Conclusion
The expression simplifies to \(\frac{4}{9} x^{-6}\), which is in the required form \(A x^m y^n\) with \(A = \frac{4}{9}\), \(m = -6\), and \(n = 0\). Thus, the final answer is \(\boxed{\frac{4}{9} x^{-6}}\).
Examples
This type of algebraic simplification is essential in fields like physics and engineering, where complex formulas involve multiple variables raised to various powers. For example, when calculating the force between two charged particles, you might deal with expressions like \(\frac{q_1 q_2}{r^2}\). Simplifying such expressions allows scientists to more easily perform sensitivity analysis, determining how a small change in distance \(r\) affects the overall force. Being able to combine powers and constants efficiently is a foundational skill for solving these real-world physical equations.