For all non-zero real numbers x and y, which of the following expressions is equivalent to… — GRE General Test

For all non-zero real numbers $x$ and $y$ , which of the following expressions is equivalent to $\frac{12x^4y^2 - 8x^2y^3}{4x^2y}$ ?

3x^2y - 2y^2

Dividing each term in the numerator by

4x^2y

yields

\frac{12x^4y^2}{4x^2y} - \frac{8x^2y^3}{4x^2y} = 3x^2y - 2y^2

, which matches the correct expression.

Split the rational algebraic expression into two separate fractions by distributing the denominator.

\frac{12x^4y^2}{4x^2y} - \frac{8x^2y^3}{4x^2y}

When dividing a polynomial by a monomial, each term of the numerator must be divided independently by the denominator.

Simplify the first term

\frac{12x^4y^2}{4x^2y}

\frac{12}{4} \cdot x^{4-2} \cdot y^{2-1} = 3x^2y

Divide the numerical coefficients and apply the exponent quotient rule

a^m / a^n = a^{m-n}

for each variable.

Simplify the second term

\frac{8x^2y^3}{4x^2y}

\frac{8}{4} \cdot x^{2-2} \cdot y^{3-1} = 2(1)y^2 = 2y^2

Divide the coefficients and apply the exponent quotient rule, noting that

x^0 = 1

for non-zero

x

Combine the simplified terms with the original subtraction operator.

3x^2y - 2y^2

Combine terms to form the final simplified expression.

Simplifying algebraic fractions by distributing a monomial denominator across numerator terms and applying exponent rules.

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