Powerful Powers

A=5^{5005}+5^{-5005}
B=5^{5005}-5^{-5005}
Using the values of A \textup{ and }B above, find the value of \left (A+B\right )\left (A-B\right ).
Source: mathcontest.olemiss.edu 6/9/2008

SOLUTION
Let n=5005. Then, A=5^{n} + 5^{-n}\textup { and }B=5^{n}-5^{-n}.
\left (A+B\right )\left (A-B\right )=A^2-B^2
=\left (5^{n}+5^{-n}\right )^2-\left (5^{n}-5^{-n}\right )^2
=\left (5^{2n}+2\cdot 5^{n}\cdot 5^{-n}+5^{-2n}\right )-\left (5^{2n}-2\cdot 5^{n}\cdot 5^{-n}+5^{-2n}\right )
=5^{2n}+2\cdot 5^{n-n}+5^{-2n}-\left (5^{2n}-2\cdot 5^{n-n}+5^{-2n}\right )
=5^{2n}+2\cdot 5^0+5^{-2n}-\left (5^{2n}-2\cdot 5^0+5^{-2n}\right )
=5^{2n}+2\cdot 1+5^{-2n}-\left (5^{2n}-2\cdot 1+5^{-2n}\right )
=5^{2n}+2+5^{-2n}-5^{2n}+2-5^{-2n}
=2+2
=4

Answer: 4.

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About mvtrinh

Retired high school math teacher.
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