Imaginary number problem

Question

Is 3i the fifth root of -243i? Justify please. 

Answer 1

(3i)(3i)(3i)(3i)(3i)

3*3*3*3*3=243

i=√-1

i*i=(√-1)(√-1)=-1

(i*i)*(i*i)*i=(-1)*(-1)*(i)=1*i=i

243i, not -243i

Answer 2

Let's look at this in 2 pieces

 

5th root of -243 is -3, not 3

 

Also

i^5 = i since

i = sq rt (-1)

i^2 = -1

i^3 = - sq rt (-1)

i^4 = 1

and

i^5 = i = sq rt (-1)

so 5th root of i^5 = i, and 5th root of i = i

 

so -3i is 5th root of -243i

 

Note I have added the "-".  Not sure if that was just a typo on your part.