Which expression is equivalent to the one below?

[tex]\[
\frac{(x^2 y)(x^4 y^3)}{x y^2}
\][/tex]

A. [tex]\(\frac{x^6 y^3}{x y^2}\)[/tex]

B. [tex]\(\frac{x^8 y^3}{x y^2}\)[/tex]

C. [tex]\(\frac{x^6 y^4}{x y^2}\)[/tex]



Answer :

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]