Mathematics, 03.07.2019 19:00 alexialoredo625
There are six different sixth roots of 64. that is, there are six complex numbers that solve x^6=64
Answers: 1
![\sqrt[n]{z}=\sqrt[n]{|z|}\Big(\cos\frac{\phi+2k\pi}{n}+i\sin\frac{\phi+2k\pi}{n}\Big)\qquad\text{for}\quad k=0,1,\ldots,n-1](/tpl/images/0047/3973/dd454.png)
![x^6=64\quad|\sqrt[6]{(\ldots)}\\\\x=\sqrt[6]{64}](/tpl/images/0047/3973/7b602.png)
we have:
![\sqrt[6]{64}=\sqrt[6]{64}\Big(\cos\frac{0+2k\pi}{6}+i\sin\frac{0+2k\pi}{6}\Big)\qquad\text{for}\quad k=0,1,\ldots,5\\\\\\ \sqrt[6]{64}=2\Big(\cos\frac{k\pi}{3}+i\sin\frac{k\pi}{3}\Big)\qquad\text{for}\quad k=0,1,\ldots,5 ](/tpl/images/0047/3973/c89c0.png)
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There are six different sixth roots of 64. that is, there are six complex numbers that solve x^6=64...
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