Let's analyze and simplify the given mathematical expression step-by-step.
The expression to simplify is:
[tex]\[
\frac{(x^2 y) (x^4 y^3)}{x y^2}
\][/tex]
1. Step 1: Simplify the numerator
Combine the terms in the numerator:
[tex]\[
(x^2 y) (x^4 y^3)
\][/tex]
To do this, use the property of exponents for multiplication: [tex]\(x^a \cdot x^b = x^{a+b}\)[/tex] and [tex]\(y^a \cdot y^b = y^{a+b}\)[/tex]:
[tex]\[
x^2 \cdot x^4 = x^{2+4} = x^6
\][/tex]
[tex]\[
y \cdot y^3 = y^{1+3} = y^4
\][/tex]
Therefore, the numerator simplifies to:
[tex]\[
x^6 y^4
\][/tex]
2. Step 2: Simplify the denominator
The denominator is already in its simplest form:
[tex]\[
x y^2
\][/tex]
which is:
[tex]\[
x^1 y^2
\][/tex]
3. Step 3: Divide the simplified numerator by the simplified denominator
Divide the exponents of [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in the numerator by those in the denominator using the property of exponents for division: [tex]\(x^a / x^b = x^{a-b}\)[/tex] and [tex]\(y^a / y^b = y^{a-b}\)[/tex]:
[tex]\[
\frac{x^6 y^4}{x y^2}
\][/tex]
Simplify the exponents:
[tex]\[
x^{6-1} = x^5
\][/tex]
[tex]\[
y^{4-2} = y^2
\][/tex]
Therefore, the simplified expression is:
[tex]\[
x^5 y^2
\][/tex]
From the provided options, none of them matches the simplified expression [tex]\(x^5 y^2\)[/tex] directly. Therefore, we should conclude that [tex]\(\frac{\left(x^2 y\right)\left(x^4 y^3\right)}{x y^2}\)[/tex] simplifies to:
[tex]\[
x^5 y^2
\][/tex]
