Of the following expressions, the one equal to a^-1b^-1a^-3-b^-3 is:

1958 AMC 12 Problem 3

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Of the following expressions, the one equal to

\dfrac{a^{-1}b^{-1}}{a^{-3}-b^{-3}}

is:

\dfrac{a^2b^2}{b^2-a^2}

\dfrac{a^2b^2}{b^3-a^3}

\dfrac{ab}{b^3-a^3}

\dfrac{a^3-b^3}{ab}

\dfrac{a^2b^2}{a-b}

Rewrite every negative power as a reciprocal

Express

a^{-3}-b^{-3}

over the common denominator

a^3b^3

We have

a^{-3}-b^{-3} =\frac{b^3-a^3}{a^3b^3}.

Hence

\frac{\frac{1}{ab}}{\frac{b^3-a^3}{a^3b^3}} =\frac{a^2b^2}{b^3-a^3}.

Thus, the correct answer is B.