For all non-zero real numbers x and y, the expression \frac{(x^2 + y^{-1})^2}{x^4} is equi… — ACT

For all non-zero real numbers $x$ and $y$ , the expression $\frac{(x^2 + y^{-1})^2}{x^4}$ is equivalent to which of the following?

1 + 2x^{-2}y^{-1} + x^{-4}y^{-2}

The correct expression is obtained by expanding the binomial in the numerator using the perfect square formula

(a + b)^2 = a^2 + 2ab + b^2

. Substituting

a = x^2

and

b = y^{-1}

yields

(x^2 + y^{-1})^2 = (x^2)^2 + 2(x^2)(y^{-1}) + (y^{-1})^2 = x^4 + 2x^2 y^{-1} + y^{-2}

. Dividing each term by the denominator

x^4

and applying the quotient rule for exponents

\frac{x^m}{x^n} = x^{m-n}

results in

1 + 2x^{-2}y^{-1} + x^{-4}y^{-2}

Expand the binomial in the numerator.

(x^2 + y^{-1})^2 = (x^2)^2 + 2(x^2)(y^{-1}) + (y^{-1})^2 = x^4 + 2x^2 y^{-1} + y^{-2}

Apply the binomial expansion formula

(a + b)^2 = a^2 + 2ab + b^2

where

a = x^2

and

b = y^{-1}

Divide each term in the expanded numerator by the denominator.

\frac{x^4}{x^4} + \frac{2x^2 y^{-1}}{x^4} + \frac{y^{-2}}{x^4}

Distribute the division over the terms in the numerator.

Simplify each fraction using exponent properties.

1 + 2x^{-2}y^{-1} + x^{-4}y^{-2}

Use the quotient rule

\frac{x^m}{x^n} = x^{m-n}

to simplify the variables.

Properties of Exponents in Algebraic Expressions

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