Negative number to fractional power - What is the solution?

Question

Hello, I'm wondering what is the solution of this kind of equation. For example (- 8) to power 1/3, will solution be (-2) or equation is not defined?

As far as I know it's possible to convert the number with power to the form of root. And cubic root of (-8) equals to (-2). But I've checked this equation in Photomath and got "not defined". Why?

I am a little bit confused so I would like to know what is the correct answer for my question. Thanks :)

Answer 1

cube root of -1/8 = cube root of -1 divided by cube root = 8 = [(-1)^1/3]/2

= one half the cube root of -1

x= the cube root of -1, then x^3 =-1

x^3+1 = 0 is a cubic equation with 3 solutions, one real 2 imaginary,

factor it

(x+1)(x^2-x + 1) = 0

set each factor = 0 and solve for x

x+1=0. x=-1

-1 is the real cube root of -1.

-1 times -1 times -1 = -1

cube root of -1/8 = one half of -1 = -1/2

two imaginary solutions involve the solution of the other factor set = 0

x^2 -x + 1 = 0

use the quadratic formula

x = 1/2 + or - (1/2)sqr(1-4) = 1/2 + or - isqr3/2

x = (1+isqr3)/2 or (1-isqr3)/2

cube roots of -1/8 = -1, (1+isqr3)/4 and (1-isqr3)/4

the last two are a conjugate pair of imaginary solutions