To simplify the expression
[tex]\[
(x^5 y^2) (x^{-6} y)
\][/tex]
we can follow several steps. Here are the details:
1. Combine the powers of [tex]\(x\)[/tex]:
When multiplying terms with the same base, you add the exponents. Therefore, we combine the powers of [tex]\(x\)[/tex]:
[tex]\[
x^5 \cdot x^{-6}
\][/tex]
Adding the exponents [tex]\(5\)[/tex] and [tex]\(-6\)[/tex] gives us:
[tex]\[
x^{5 + (-6)} = x^{-1}
\][/tex]
2. Combine the powers of [tex]\(y\)[/tex]:
Similarly, we combine the powers of [tex]\(y\)[/tex]:
[tex]\[
y^2 \cdot y
\][/tex]
Adding the exponents [tex]\(2\)[/tex] and [tex]\(1\)[/tex] gives us:
[tex]\[
y^{2 + 1} = y^3
\][/tex]
3. Multiply the simplified parts:
Putting it all together, we have:
[tex]\[
x^{-1} y^3
\][/tex]
4. Express in the given choices format:
[tex]\(\displaystyle x^{-1}\)[/tex] can be rewritten as [tex]\(\displaystyle \frac{1}{x}\)[/tex]. Thus, the expression [tex]\(\displaystyle x^{-1} y^3\)[/tex] is equivalent to:
[tex]\[
\frac{y^3}{x}
\][/tex]
Hence, the correct simplification of the expression [tex]\(\left(x^5 y^2\right)\left(x^{-6} y\right)\)[/tex] is:
[tex]\[
\boxed{\frac{y^3}{x}}
\][/tex]
