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
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