How do you formally determine the sign of a root?

Question

While reading about parametrization, I came across the example of the cuspidal cubic which is defined by $y^2=x^3$. It's stated that an equivalent function is $\vec{f}(t) = \begin{bmatrix} t^2 \\ t^3 \end{bmatrix}$. I understand that we have $x=t^2 \implies y^2=(t^2)^3=t^6 \implies \sqrt{y^2} = y = \sqrt{t^6} = t^3$. However, I'm confused about the sign of $y^2 = t^6$. Say we have $t = -2$. Then $y^2 = (-2)^6 = 64 \implies y = \pm 8$. As I understand, each value of $t$ should be paired with only one value of $y$, so should $y$ equal $8$ or $-8$? In general, how do you deal with the nuances of the signs of root operations?

Answer 1

The ambiguity in the sign of the root comes from the ambiguity you have when expressing $y$ in terms of $x$: for each value of $x$, there are two points that lay on the curve $y^2-x^3=0$, namely $(x,y)$ and $(x,-y)$. When the author takes the parametrization $(t^2,t^3)$ he has implicitly already chosen a sign of the square root. Note that $(t^2,-t^3)$ is another perfectly valid parametrization. The important thing is that, once you’ve chosen a sign of the root, the parametrization gives you a bijection between the values of the parameter and the points on the curve.