How can I find the point of intersection between these functions?

Question

Find the point of intersection between $f(x) = x^3$ and $g(x) = x^{1/3}$

Once I equal each equation to each other, I could factor out the $x$ but the exponent 1/3 is confusing me.

Thank you!

Answer 1

Set $f = g$, $$x^3 = x^{\frac{1}{3}} \Rightarrow (x^3)^3 = (x^{\frac{1}{3}})^3 \Rightarrow x^9 = x$$ $$\Rightarrow x^9 -x = x(x^8-1) = x(x-1)(x+1)(x^2+1)(x^4+1) = 0$$

So your roots are $$x = 0$$ $$x =\pm 1$$