root(6)(-64)=?Please,give all the possible answers.

1 Answer
May 23, 2018

See beow

Explanation:

Calculate root(6)(-64) means you have to find a real number x such that x^6=-64. Such number doesnt exist because if it were positive, then never will get a negative number as product, if it were negative, then

(-x)·(-x)·(-x)·(-x)·(-x)·(-x)= positive number (there are an even number of factors (6) and never will get -64)

In summary that root(6)(-64) has no real solutions. There is no number x such that x^6=-64

But in complex set of numbers there are 6 solutions

First put -64 in polar form which is 64_180

Then the six solutions r_i from i=0 to i=5 are

r_0=root(6)64_(180/6)=2_30
r_1=root(6)64_((180+360)/6)=2_90
r_2=2_((180+720)/6)=2_150
r_3=2_((180+1080)/6)=2_210
r_4=2_270
r_5=2_330

Who are these numbers?

r_0=2(cos30+isin30)=sqrt3+i
r_1=2i
r_2=-sqrt3+i
r_3=-sqrt3-i
r_4=-2i
r_5=sqrt3-i