1952 AHSME Problems/Problem 3

Problem

The expression $a^3-a^{-3}$ equals:

$\textbf{(A) \ }\left(a-\frac{1}{a}\right)\left(a^2+1+\frac{1}{a^2}\right) \qquad \textbf{(B) \ }\left(\frac{1}{a}-a\right)\left(a^2-1+\frac{1}{a^2}\right) \qquad \textbf{(C) \ }\left(a-\frac{1}{a}\right)\left(a^2-2+\frac{1}{a^2}\right) \qquad$ $\textbf{(D) \ }\left(\frac{1}{a}-a\right)\left(\frac{1}{a^2}+1+a^2\right) \qquad \textbf{(E) \ }\text{none of these}$

Solution

Recall that $x^3-y^3=(x-y)(x^2+xy+y^2)$ . Letting $a=x$ and $a^{-1}=y$ , we find that $a^3-a^{-3}= \boxed{\textbf{(A)}\ \left(a-\frac{1}{a}\right)\left(a^2+1+\frac{1}{a^2}\right)}$ .

1952 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 2	Followed by Problem 4
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1952 AHSME Problems/Problem 3

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