Question

Why doesn't a cube root have two answers

Answer 1

A cube root has THREE answers !

Unfortunately, two of them are 'complex' numbers, which involve
'imaginary' numbers', and those involve the square root of -1 . 
In Middle School, you're not equipped to go there yet. 

Just hang on for a few more years.  You'll learn all kinds of
beautiful things in math, more than you can imagine now.
You'll learn that whatever number the root is, there are
THAT many answers.

Like for example, if you want the 7th root of a number,
there are seven of them !