# Quickie #15

• Jan 23rd 2007, 09:26 AM
Soroban
Quickie #15

Simplify: . $\left(1 + 2^{-\frac{1}{32}}\right)\left(1 + 2^{-\frac{1}{16}}\right)\left(1 + 2^{-\frac{1}{8}}\right)\left(1 + 2^{-\frac{1}{4}}\right)\left(1 + 2^{-\frac{1}{2}}\right)$

Edit: 34 views and no soltuion yet?
• Jan 25th 2007, 08:32 AM
Soroban
Finally found one that stumped all of you?

Let: . $X \;=\;\left(1 + 2^{-\frac{1}{32}}\right)\left(1 + 2^{-\frac{1}{16}}\right)\left(1 + 2^{-\frac{1}{8}}\right)\left(1 + 2^{-\frac{1}{4}}\right)\left(1 + 2^{-\frac{1}{2}}\right)$

Multiply both sides by $\left(1 - 2^{-\frac{1}{32}}\right)$

$\left(1 - 2^{-\frac{1}{32}}\right)X \;=\;\underbrace{\left(1 - 2^{-\frac{1}{32}}\right)\left(1 + 2^{-\frac{1}{32}}\right)}\left(1 + 2^{-\frac{1}{16}}\right)$ $\left(1 + 2^{-\frac{1}{8}}\right)\left(1 + 2^{-\frac{1}{4}}\right)\left(1 + 2^{-\frac{1}{2}}\right)$

. . . . . . . . . . . . . . $= \;\underbrace{\left(1 - 2^{-\frac{1}{16}}\right) \left(1 + 2^{-\frac{1}{16}}\right)}\left(1 + 2^{-\frac{1}{8}}\right)\left(1 + 2^{-\frac{1}{4}}\right)\left(1 + 2^{-\frac{1}{2}}\right)$

. . . . . . . . . . . . . . . . . . $= \;\underbrace{\left(1 - 2^{-\frac{1}{8}}\right)\left(1 + 2^{-\frac{1}{8}}\right)}\left(1 + 2^{-\frac{1}{4}}\right)\left(1 + 2^{-\frac{1}{2}}\right)$

. . . . . . . . . . . . . . . . . . . . . . $=\;\underbrace{\left(1 - 2^{-\frac{1}{4}}\right)\left(1 + 2^{-\frac{1}{4}}\right)}\left(1 + 2^{-\frac{1}{2}}\right)$

. . . . . . . . . . . . . . . . . . . . . . . . . . $= \;\underbrace{\left(1 - 2^{-\frac{1}{2}}\right)\left(1 + 2^{-\frac{1}{2}}\right)}$

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $= \;1 - 2^{-1}$

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $=\;\frac{1}{2}$

Therefore: . $X \;= \;\frac{\frac{1}{2}}{1 - 2^{-\frac{1}{32}}} \;=\;\frac{1}{2 - 2^{\frac{31}{32}}}

$

• Jan 25th 2007, 10:09 AM
anthmoo
Thankyou! I was waiting forever for someone to do this! :D
• Jan 25th 2007, 02:46 PM
topsquark
Quote:

Originally Posted by Soroban
Finally found one that stumped all of you?

Of course not! My calculator came up with 23.334023182557 a long time ago! :p

-Dan