Maths Olympiad Prep

Library / /81 of 520

Algebra Difficulty 2.1 Junior Find the answer

The fraction a−4−b−4a−2−b−2\frac{a^{-4}-b^{-4}}{a^{-2}-b^{-2}} is equal to:
(A) a−6−b−6\textbf{(A)}\ a^{-6}-b^{-6}(B) a−2−b−2\textbf{(B)}\ a^{-2}-b^{-2}(C) a−2+b−2(D) a2+b2\textbf{(C)}\ a^{-2}+b^{-2}\\ \textbf{(D)}\ a^2+b^2(E) a2−b2\textbf{(E)}\ a^2-b^2

Multiple choice: answer with the letter of the option you want.

Solution

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

Want a route through all this instead of an archive? The track puts 2,000 problems in a working order, from AMC 10 level to the IMO shortlist.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.