$(\frac{z-z^{-1}}{2i})^4 $ simplified
You asked:
Simplify the expression: \({\left( \frac{z - {z}^{-1}}{2 i} \right)}^{4}\)
MathBot Answer:
Simplified:
\[ {\left( \frac{z - {z}^{-1}}{2 i} \right)}^{4} = \frac{\left(z^{2} - 1\right)^{4}}{16 z^{4}} \]
Expanded:
\[ {\left( \frac{z - {z}^{-1}}{2 i} \right)}^{4} = \frac{z^{4}}{16} - \frac{z^{2}}{4} + \frac{3}{8} - \frac{1}{4 z^{2}} + \frac{1}{16 z^{4}} \]
Factored:
\[ {\left( \frac{z - {z}^{-1}}{2 i} \right)}^{4} = \frac{\left(z - 1\right)^{4} \left(z + 1\right)^{4}}{16 z^{4}} \]