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1955 AHSME Problems/Problem 13

Revision as of 12:21, 1 August 2020 by Angrybird029 (talk | contribs) (Created page with "== Problem 13== The fraction <math>\frac{a^{-4}-b^{-4}}{a^{-2}-b^{-2}}</math> is equal to: <math> \textbf{(A)}\ a^{-6}-b^{-6}\qquad\textbf{(B)}\ a^{-2}-b^{-2}\qquad\textbf{...")
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Problem 13

The fraction $\frac{a^{-4}-b^{-4}}{a^{-2}-b^{-2}}$ is equal to:

$\textbf{(A)}\ a^{-6}-b^{-6}\qquad\textbf{(B)}\ a^{-2}-b^{-2}\qquad\textbf{(C)}\ a^{-2}+b^{-2}\\ \textbf{(D)}\ a^2+b^2\qquad\textbf{(E)}\ a^2-b^2$

Solution

By the difference of squares property, $a^{-4} - b^{-4}$ is equivalent to $(a^{-2} + b^{-2})(a^{-2} - b^{-2})$. This means the fraction is now equal to $\frac{(a^{-2} + b^{-2})(a^{-2} - b^{-2})}{a^{-2} - b^{-2}}$, which simplifies to $\textbf{(C)}\ a^{-2}+b^{-2}$

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