When is it OK to take powers of the equation of a curve?

Question

Isnt raising curve equations to powers "dangerous" in general? Take

$$ x=y $$

if you square that, you get $$x^2 = y^2$$

The new equation contains the points described by $ x=y $, and $ x=-y $ combined, ie, 2 lines, slopes 1 & -1, unlike the original equation that only described a single line.

I'm self-studying calculus currently, reached arclengths, and I find that I have to take powers of curve equations sometimes or I CANNOT solve a problem. Im always worried about unwanted solutions. Is there some easy way of telling when it's ok to take powers?

The particular problem I'm struggling with currently is one where I have to get an equation in $x$ & $y$ for this parametric equation for a curve ($k$ is a positive integer):

$$ x = a.cos^k (t) $$

$$ y = a.sin^k (t) $$

Is it valid to take the both sides to the power $2/k$ here? That would certainly help me solve this problem, but I'm not comfortable doing that, because I have no idea what kind of extra solutions will spawn, Im struggling to even imagine what it could be doing to the solution.

I'd be so grateful if anyone could help clear up this recurring doubt I'm having :⁠'⁠(.

Answer 1

In short, no, unless you have constraints to eliminate unwanted solution. For example if you have the constraint $x > 0$ then it's find to take the square root of both sides of $x^2 = y^2$, which is $x = |y|$. You problem with the parametric curve can be solved though. Notice that $sin^2(t)$ and $cos^2(t)$ are non-negative, then: $$(x^2)^{1/k}+(y^2)^{1/k} = (a^2)^{1/k}$$ Please remember that $(x^2)^{1/k}$ is not the same as $x^{2/k}$ for $x < 0$. You can also use the absolute value function: $$|x|^{2/k}+|y|^{2/k} = |a|^{2/k}$$

Answer 2

After you square, everything becomes positive, so then you can take the $k$th root without any problem. The equation $x=y$ implies $x^2=y^2$ though of course one cannot reverse the implication.