1958 AHSME Problems/Problem 3

Problem

Of the following expressions the one equal to $\frac{a^{-1}b^{-1}}{a^{-3} - b^{-3}}$ is:

$\textbf{(A)}\ \frac{a^2b^2}{b^2 - a^2}\qquad \textbf{(B)}\ \frac{a^2b^2}{b^3 - a^3}\qquad \textbf{(C)}\ \frac{ab}{b^3 - a^3}\qquad \textbf{(D)}\ \frac{a^3 - b^3}{ab}\qquad \textbf{(E)}\ \frac{a^2b^2}{a - b}$

Solution

$\frac{a^{-1}b^{-1}}{a^{-3} - b^{-3}} = \frac{\frac{1}{ab}}{\frac{1}{a^{3}}-\frac{1}{b^{3}}} = \frac{\frac{1}{ab}}{\frac{1}{a^{3}}-\frac{1}{b^{3}}}\cdot\frac{a^{3}b^{3}}{a^{3}b^{3}} = \frac{a^{2}b^{2}}{b^{3}-a^{3}}$ , $\boxed{\text{B}}$ .

1958 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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1958 AHSME Problems/Problem 3

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