indices

• January 28th 2009, 05:12 AM
requal
indices
if $a=2^x+2^{-x}$, and $b=2^x-2^{-x}$, find an expression for $a^2+b^2$
• January 28th 2009, 05:26 AM
Soroban
Hello, requal!

Did you do the algebra?

Quote:

If $a\,=\,2^x+2^{-x}$ and $b\,=\,2^x-2^{-x}$, find an expression for $a^2+b^2$

. . $a^2 \:=\:\left(2^x + 2^{-x}\right)^2 \:=\:(2^x)^2 + 2(2^x)(2^{-x}) + (2^{-x})^2 \:=\:2^{2x} + 2 + 2^{-2x}$ .[1]

. . $b^2 \:=\:\left(2^x-2^{-x}\right)^2 \:=\:(2^x)^2 - 2(2^x)(2^{-x}) + (2^{-x})^2 \:=\:2^{2x} - 2 + 2^{-2x}$ .[2]

Add [1] and [2]: . $a^2+b^2 \;=\;2\cdot2^{2x} + 2\cdot2^{-2x} \;=\;2^{2x+1} + 2^{-2x+1}$