Answer
The number of intersections between the graphs of $y=x^n$ and $y=a$ give the number of roots
Work Step by Step
We have to study the equation:
$$\begin{align*}x=\sqrt[n]a\end{align*}\tag1$$
for $n$ even.
Equation $(1)$ can be written:
$$x^n=a.$$
We are given the graph of the function $f(x)=x^n$, where $n$ is even. We have to study the intersection between the graph of the function $f$ and the line $y=a$.
Case 1: $a<0$
There is no intersection between the two graphs, therefore the equation has no real solution.
Case 2: $a=0$
There is one intersection for $x=0$, so the equation has one solution.
Case 3: $a>0$
There are two intersections, so there are two solutions $\sqrt[n]a$ and $-\sqrt[n]a$.
The graph justifies the conclusions from the Key Concept box for $n$ positive even integer.
