For all non-zero real numbers p and q, which of the following expressions is equivalent to… — ACT

For all non-zero real numbers

p

and

q

, which of the following expressions is equivalent to

\frac{(p^2 + q^{-1})^2 - p^4}{q^{-2}}?

2p^2 q + 1

Expanding the binomial in the numerator yields

p^4 + 2p^2 q^{-1} + q^{-2}

. After subtracting

p^4

, the numerator is left as

2p^2 q^{-1} + q^{-2}

. Dividing this numerator term-by-term by the denominator

q^{-2}

gives

\frac{2p^2 q^{-1}}{q^{-2}} + \frac{q^{-2}}{q^{-2}}

. Applying the quotient property of exponents to each term yields

2p^2 q^{(-1) - (-2)} + 1 = 2p^2 q^1 + 1 = 2p^2 q + 1

Expand the binomial in the numerator using the perfect square identity:

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

(p^2 + q^{-1})^2 = (p^2)^2 + 2(p^2)(q^{-1}) + (q^{-1})^2 = p^4 + 2p^2 q^{-1} + q^{-2}

To begin simplifying the numerator, we must resolve the exponent outside the parentheses.

Subtract

p^4

from the expanded numerator expression.

(p^4 + 2p^2 q^{-1} + q^{-2}) - p^4 = 2p^2 q^{-1} + q^{-2}

This simplifies the numerator by combining the

p^4

terms.

Divide each term in the simplified numerator by the denominator

q^{-2}

\frac{2p^2 q^{-1}}{q^{-2}} + \frac{q^{-2}}{q^{-2}} = 2p^2 q^{-1 - (-2)} + q^{-2 - (-2)}

To divide a polynomial by a monomial, we distribute the division to each term of the polynomial.

Apply the quotient rule of exponents,

\frac{x^a}{x^b} = x^{a-b}

, and simplify the terms.

2p^2 q^1 + q^0 = 2p^2 q + 1

Subtracting the exponents of

q

in each term simplifies the division to its final form.

Properties of Exponents in Algebraic Expressions

Question