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# Complex Numbers

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What is the value of (1 + i)^5 - (1 - i)^5?

Aug 14, 2021

#1
+26228
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What is the value of $$(1 + i)^5 - (1 - i)^5$$?

$$\begin{array}{|rcll|} \hline (1+i)^5-(1-i)^5 &=& \left((1+i)^2\right)^2(1+i)-\left((1-i)^2\right)^2(i-1) \\ &=& \left(1+2i+i^2\right)^2(1+i)-\left(1-2i+i^2\right)^2(1-i) \\ &=& \left(1+2i-1\right)^2(1+i)-\left(1-2i-1\right)^2(1-i) \\ &=& \left(2i\right)^2(1+i)-\left(-2i\right)^2(1-i) \\ &=& 4i^2(1+i)-4i^2(1-i) \quad | \quad i^2 =-1 \\ &=& -4(1+i)+4(1-i) \\ &=& -4-4i+4-4i \\ \mathbf{(1+i)^5-(1-i)^5} &=& \mathbf{-8i} \\ \hline \end{array}$$

Aug 14, 2021