1951 AMC 12 Problem 20

Attempt Problem 20 of the 1951 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1951 AMC 12 solutions, or check the answer key.

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20.

When simplified and expressed with negative exponents, the expression

(x+y)^{-1}(x^{-1}+y^{-1})

is equal to:

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

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

x^{-1}y^{-1}

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

\dfrac{1}{x^{-1}y^{-1}}

Rewrite

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

as one fraction

The factor

x+y

cancels after combining the reciprocals

For nonzero

x,y

with

x+y\ne0,

\begin{aligned} &(x+y)^{-1}(x^{-1}+y^{-1})\\ &\quad=\frac{1}{x+y}\left(\frac{x+y}{xy}\right)\\ &\quad=\frac1{xy}=x^{-1}y^{-1}. \end{aligned}

Thus, the correct answer is C.

Problem 20 in other years