Roots of unity on the axes

The only roots of unity lying on the axes of the complex plane are $1,i,-1,-i$ .
We can check them in turn:
$1^{14}=1$
$i^{14}=-1$
$(-1)^{14}=1$
$(-i)^{14}=-1$
So there only two satisfying the conditions: $1$ and $-1$

The roots are of the form $cos\left(\dfrac{2πk}{14}\right)+isin\left(\dfrac{2πk}{14}\right)$ for $1 \leq k \leq 14$

Now, if the root lies on the axes, then either the abscissa (x coordinate) or the ordinate (y coordinate) is zero.

It is not hard to see that in the range $1 \leq k \leq 14$ , the only values for which $isin\left(\dfrac{2πk}{14}\right)=0$ are $k=7$ and $14$ which gives the roots $-1$ and $1$ , respectively.

For $cos\left(\dfrac{2πk}{14}\right)=0$ , we find that there is no integer value of $k$ satisfying this equation.

Hence, there are only two values on the axes ( $-1$ and $1$ ) and thus the answer is $\boxed{2}$