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Compute (1 + sqrt(3)*i)^10.

 May 10, 2020
 #1
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      Compute (1 + sqrt(3)*i)^10.                   

 

243 i^10 + 810 sqrt(3) i^9 + 3645 i^8 + 3240 sqrt(3) i^7 + 5670 i^6 + 2268 sqrt(3) i^5 + 1890 i^4 + 360 sqrt(3) i^3 + 135 i^2 + 10 sqrt(3) i + 1    (11 terms)

 May 10, 2020
 #2
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Compute

\((1 + \sqrt{3}*i)^{10}\).

 

\(\begin{array}{|rcll|} \hline \mathbf{ (1 + \sqrt{3}*i)^{10} } &=& \left(\sqrt{1^2 + (\sqrt{3})^2}*e^{i*\arctan\left(\frac{\sqrt{3}}{1}\right)} \right)^{10}\\ &=& \left(2*e^{i*\arctan\left(\sqrt{3}\right)} \right)^{10}\\ &=& \left(2*e^{i*\frac{\pi}{3}} \right)^{10}\\ &=& 2^{10}*e^{i*\frac{10}{3}\pi}\\ &=& 2^{10}*\left(\cos(\frac{10}{3}\pi)+i*\sin(\frac{10}{3}\pi) \right) \\ &=& 2^{10}*\left(-0.5+i* \left(-\dfrac{\sqrt{3}}{2} \right) \right) \\ &=& 1024*\left(-0.5+i* \left(-\dfrac{\sqrt{3}}{2} \right) \right) \\ &=& 1024*(-0.5)+i* 1024\left(-\dfrac{\sqrt{3}}{2} \right) \\ &=& -512-i* 512\sqrt{3} \\ \mathbf{ (1 + \sqrt{3}*i)^{10} }&=& \mathbf{-512(1+ \sqrt{3}*i)} \\ \hline \end{array} \)

 

laugh

 May 11, 2020

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