Sure! To expand the expression [tex]\(2 x^4 y^4\left(x^{-1} y-2 x^2 y^{-2}\right)\)[/tex] using the distributive property, follow these steps:
1. Distribute the terms inside the parentheses to each term outside:
[tex]\[
2 x^4 y^4 \cdot x^{-1} y - 2 x^4 y^4 \cdot 2 x^2 y^{-2}
\][/tex]
2. Simplify each term individually:
- For the first term [tex]\(2 x^4 y^4 \cdot x^{-1} y\)[/tex]:
[tex]\[
2 x^4 y^4 \cdot x^{-1} y = 2 \left(x^4 \cdot x^{-1}\right) \left(y^4 \cdot y\right)
\][/tex]
Using the law of exponents where [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex]:
[tex]\[
2 \left(x^{4-1}\right) \left(y^{4+1}\right) = 2 x^3 y^5
\][/tex]
- For the second term [tex]\(2 x^4 y^4 \cdot 2 x^2 y^{-2}\)[/tex]:
[tex]\[
2 x^4 y^4 \cdot 2 x^2 y^{-2} = 2 \cdot 2 \left(x^4 \cdot x^2\right) \left(y^4 \cdot y^{-2}\right)
\][/tex]
Simplify the constants:
[tex]\[
4 \left(x^{4+2}\right) \left(y^{4-2}\right) = 4 x^6 y^2
\][/tex]
3. Combine the simplified terms:
Collect the results:
[tex]\[
2 x^3 y^5 - 4 x^6 y^2
\][/tex]
Thus, the expanded expression is:
[tex]\[
-4 x^6 y^2 + 2 x^3 y^5
\][/tex]
That’s the final expanded form of the given expression using the distributive property.
