What is \((i-i^{-1})^{-1}\)?

I honestly just don't know how to do this because it just looks so confusing. Please help! All help is greatly appreciated! :D

lokiisnotdead Sep 16, 2020

#1**+1 **

Remember that raising a number to the power of -1 "flips" it so the numerator becomes the denominator and vice versa.

Like this: ( a / b )^{-1} = b / a and ( a )^{-1} = ( a / 1 )^{-1} = 1 / a

\((i-i^{-1})^{-1}\\~\\ =\quad(i-\frac1i)^{-1}\)

Get a common denominator between \(i\) and \(\frac1i\) by multiplying the first term by \(\frac{i}{i}\)

\(=\quad(i\cdot\frac{i}{i}-\frac1i)^{-1}\\~\\ =\quad(\frac{i^2}{i}-\frac1i)^{-1}\\~\\ =\quad(\frac{i^2-1}{i})^{-1}\)

Replace \(i^2\) with -1

\(=\quad(\frac{-1-1}{i})^{-1}\\~\\ =\quad(\frac{-2}{i})^{-1}\\~\\ =\quad\frac{i}{-2}\\~\\ =\quad-\frac{1}{2}i\)

Check: https://www.wolframalpha.com/input/?i=%28i-i%5E-1%29%5E-1

hectictar Sep 16, 2020