Question
Which expression is equivalent to \( \left(\frac{x^{-4} y}{x^{-9} y^{5}}\right)^{-2} \) ? Assume \( x \neq 0, y \neq 0 \)
Ask by Hodges Reed. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The equivalent expression is \( \frac{y^{8}}{x^{10}} \).
Solution
1. Start with the given expression:
\[
\left(\frac{x^{-4} y}{x^{-9} y^{5}}\right)^{-2}
\]
2. Simplify the fraction inside the parentheses. Using the rule \(\frac{a^m}{a^n} = a^{m-n}\):
- For \(x\):
\[
x^{-4} \div x^{-9} = x^{-4 - (-9)} = x^{5}
\]
- For \(y\):
\[
y^{1} \div y^{5} = y^{1-5} = y^{-4}
\]
So, the fraction simplifies to:
\[
x^{5} y^{-4}
\]
3. Now raise the result to the power of \(-2\) using the power rule \((a^m)^n = a^{mn}\):
\[
\left(x^{5} y^{-4}\right)^{-2} = x^{5(-2)} y^{-4(-2)} = x^{-10} y^{8}
\]
4. Therefore, the equivalent expression is:
\[
\frac{y^{8}}{x^{10}}
\]
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Simplify this solution The Deep Dive
To simplify the expression \( \left(\frac{x^{-4} y}{x^{-9} y^{5}}\right)^{-2} \), we first simplify the fraction inside the parentheses: \[ \frac{x^{-4} y}{x^{-9} y^{5}} = \frac{y}{y^5} \cdot \frac{x^{-4}}{x^{-9}} = y^{1-5} \cdot x^{-4 - (-9)} = y^{-4} \cdot x^{5} = \frac{x^5}{y^4} \] Now we raise this to the power of -2: \[ \left(\frac{x^5}{y^4}\right)^{-2} = \frac{y^{8}}{x^{10}} = \frac{y^{8}}{x^{10}} \] So the equivalent expression is: \[ \frac{y^{8}}{x^{10}} \]
