Question

Find all the values of z =64 1/3

Answer 1

Answer:

Step-by-step explanation:

5555555555

Answer 2

Answer:

Concept:

n th root of a complex number is found by using demovire's theorem

$(cos\theta+i\:sin\theta)^n=cos\:n\theta+i\:sn\:n\theta$

$Now,\\\\z=(64)^{\frac{1}{3}}\\\\z=(64(1))^{\frac{1}{3}}\\\\z=(64[cos0+i\:sin0])^{\frac{1}{3}}\\\\z=(64)^{\frac{1}{3}}[cos2k\pi+i\:sin2k\pi]^{\frac{1}{3}}\\\\z=(4^3)^{\frac{1}{3}}[cos\frac{2k\pi}{3}+i\:sin\frac{2k\pi}{3}]\:\:\:k=0, 1, 2\\\\z=4[cos\frac{2k\pi}{3}+i\:sin\frac{2k\pi}{3}]\:\:\:k=0, 1, 2$

$The\:values\:are\\\\when\:k=0, \:\:z=4[cos0+i\:sin0]=4\\\\when\:k=1, \:\:z=4[cos\frac{2\pi}{3}+i\:sin\frac{2\pi}{3}]\\\\when\:k=2, \:\:z=4[cos\frac{4\pi}{3}+i\:sin\frac{4\pi}{3}]$