1951 AHSME Problems/Problem 20

Problem

When simplified and expressed with negative exponents, the expression $(x \plus{} y)^{ \minus{} 1}(x^{ \minus{} 1} \plus{} y^{ \minus{} 1})$ (Error compiling LaTeX. Unknown error_msg) is equal to:

$\textbf{(A)}\ x^{ \minus{} 2} \plus{} 2x^{ \minus{} 1}y^{ \minus{} 1} \plus{} y^{ \minus{} 2} \qquad\textbf{(B)}\ x^{ \minus{} 2} \plus{} 2^{ \minus{} 1}x^{ \minus{} 1}y^{ \minus{} 1} \plus{} y^{ \minus{} 2} \qquad\textbf{(C)}\ x^{ \minus{} 1}y^{ \minus{} 1}$ (Error compiling LaTeX. Unknown error_msg) $\textbf{(D)}\ x^{ \minus{} 2} \plus{} y^{ \minus{} 2} \qquad\textbf{(E)}\ \frac {1}{x^{ \minus{} 1}y^{ \minus{} 1}}$ (Error compiling LaTeX. Unknown error_msg)

Solution

Note that $(x + y)^{-1}(x^{-1} + y^{-1}) = \dfrac{1}{x + y}\cdot\left(\dfrac{1}{x} + \dfrac{1}{y}\right) = \dfrac{1}{x + y}\cdot\dfrac{x + y}{xy} = \dfrac{1}{xy} = x^{-1}y^{-1}$ . The answer is $\textbf{(C)}$ .

1951 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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1951 AHSME Problems/Problem 20

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