複素数の計算

■問題

次の複素数の計算をせよ．

${\displaystyle {\left(1+\sqrt{3}i\right)}^7}$

■答

${\displaystyle 64\left(1+\sqrt{3}i\right)}$

■解説

${\displaystyle {\left(1+\sqrt{3}i\right)}^7}$${\displaystyle ={\left\{{\left(1+\sqrt{3}i\right)}^2\right\}}^2{\left(1+\sqrt{3}i\right)}^3}$

${\displaystyle ={\left(-2+2\sqrt{3}i\right)}^2\left(-8\right)}$

${\displaystyle ={\left\{-2\left(1-2\sqrt{3}i\right)\right\}}^2\left(-8\right)}$

${\displaystyle =4\left(-8\right){\left(1-\sqrt{3}i\right)}^2}$

${\displaystyle =-32\left(1-2\sqrt{3}i+3i^2\right)}$

${\displaystyle =-32\left(1-2\sqrt{3}i-3\right)}$

${\displaystyle =-32\left(-2-2\sqrt{3}i\right)}$

${\displaystyle =64\left(1+\sqrt{3}i\right)}$

■別解

${\displaystyle 1+\sqrt{3}i}$ の極形式はこの問題より

${\displaystyle 1+\sqrt{3}i}$${\displaystyle =2\left(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right)}$

となる．

ド・モアブルの定理を用いると

${\displaystyle {\left(1+\sqrt{3}i\right)}^7}$${\displaystyle ={\left\{2\left(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right)\right\}}^7}$

${\displaystyle =2^7\left(\cos \frac{7\pi }{3}+i\sin \frac{7\pi }{3}\right)}$

${\displaystyle =128\left\{\cos \left(2\pi +\frac{\pi }{3}\right)+i\sin \left(2\pi +\frac{\pi }{3}\right)\right\}}$

${\displaystyle =128\left(\cos \frac{\pi }{3}+i\sin \frac{\pi }{3}\right)}$

${\displaystyle =128\left(\frac{1}{2}+\frac{\sqrt{3}}{2}i\right)}$

${\displaystyle =64\left(1+\sqrt{3}i\right)}$

 

 

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最終更新日： 2023年2月25日