Answer
See explanations below.
Work Step by Step
Step 1. Write $1$ in polar form: $1=1+0i=cos0+i\cdot sin0$
Step 2. The $n$ distinct root of $1$ are: $1^{\frac{1}{n}}=cos\frac{0+2k\pi}{n}+i\cdot sin\frac{0+2k\pi}{n}
=cos\frac{2k\pi}{n}+i\cdot sin\frac{2k\pi}{n}$ where $k=0,1,2,...n-1$
Step 3. With the definition of $w$, use De Moivre's Theorem, we have $w^k=cos\frac{2k\pi}{n}+i\cdot sin\frac{2k\pi}{n}$
Step 4. Compare the results from Steps 2 and 3, we conclude that the $n$ distinct roots of $1$ are $1,w,w^2,w^3,..., w^{n-1}$
