Certainly! Let's simplify the expression [tex]\[(xy^2)^{-4}\][/tex]
### Step-by-Step Solution:
1. Original Expression:
[tex]\[
(xy^2)^{-4}
\][/tex]
2. Distribute the Exponent:
The negative exponent [tex]\(-4\)[/tex] applies to both [tex]\(x\)[/tex] and [tex]\(y^2\)[/tex] inside the parentheses. According to the property [tex]\((ab)^n = a^n b^n\)[/tex], we can distribute the exponent:
[tex]\[
(xy^2)^{-4} = x^{-4} \cdot (y^2)^{-4}
\][/tex]
3. Simplify [tex]\( (y^2)^{-4} \)[/tex]:
Now, let's simplify [tex]\((y^2)^{-4}\)[/tex]. By using the power rule [tex]\((a^m)^n = a^{mn}\)[/tex], we have:
[tex]\[
(y^2)^{-4} = y^{2 \cdot (-4)} = y^{-8}
\][/tex]
4. Combine and Express with Positive Exponents:
Combine the results:
[tex]\[
x^{-4} \cdot y^{-8}
\][/tex]
A negative exponent indicates a reciprocal, so [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex]. Applying this property, we get:
[tex]\[
x^{-4} \cdot y^{-8} = \frac{1}{x^4} \cdot \frac{1}{y^8}
\][/tex]
5. Final Expression:
Combine the fractions:
[tex]\[
\frac{1}{x^4} \cdot \frac{1}{y^8} = \frac{1}{x^4 y^8}
\][/tex]
### Simplified Expression:
[tex]\[
\left(x y^2\right)^{-4} = \frac{1}{x^4 y^8}
\][/tex]
This is the simplified form of the given expression [tex]\((xy^2)^{-4}\)[/tex].
