Certainly! Let's solve the given expression step-by-step using the given values \( x = 3 \) and \( y = -4 \).
The expression given is:
[tex]\[ \frac{x y^{-2}}{3 x^2 y^{-4}} \][/tex]
### Step-by-Step Solution:
1. Simplify the expression:
[tex]\[
\frac{x y^{-2}}{3 x^2 y^{-4}}
\][/tex]
Using properties of exponents: \( y^{-a} = \frac{1}{y^a} \), we can rewrite the expression.
2. Combine the exponents:
[tex]\[
= \frac{x y^{-2}}{3 x^2 \cdot y^{-4}}
= \frac{x \cdot \frac{1}{y^2}}{3 x^2 \cdot \frac{1}{y^4}}
= \frac{x \cdot \frac{1}{y^2}}{3 x^2 \cdot \frac{1}{y^4}}
= \frac{x}{y^2} \cdot \frac{y^4}{3 x^2}
= \frac{x \cdot y^4}{y^2 \cdot 3 x^2}
\][/tex]
3. Simplify further by canceling out common terms:
[tex]\[
= \frac{x \cdot y^4}{3 x^2 \cdot y^2}
= \frac{x \cdot y^2 \cdot y^2}{3 x^2 \cdot y^2}
= \frac{x \cdot y^2}{3 x^2}
= \frac{y^2}{3 x}
\][/tex]
So the expression simplifies to:
[tex]\[
\frac{1}{3} \cdot x^{-1} \cdot y^2
\][/tex]
4. Substitute the given values \( x = 3 \) and \( y = -4 \):
[tex]\[
\left( \frac{1}{3} \right) \cdot 3^{-1} \cdot (-4)^2
\][/tex]
5. Evaluate the individual terms:
[tex]\[
= \left( \frac{1}{3} \right) \cdot \left( \frac{1}{3} \right) \cdot 16
\][/tex]
Since \(3^{-1} =\frac{1}{3}\) and \( (-4)^2 = 16 \)
6. Multiply them together:
[tex]\[
= \left( \frac{1}{3} \right) \cdot \left( \frac{1}{3} \right) \cdot 16
= \left( \frac{1}{3} \right) \cdot \left( \frac{16}{3} \right)
= \frac{16}{9}
\][/tex]
Thus, the final evaluated result is:
[tex]\[ \frac{16}{9} \approx 1.7777777777777777 \][/tex]
