Why the graph of $y=x^{5/2}$ lies between $y=x^2$ and $y=x^3$?

Question

Why the graph of $y=x^{5/2}$ lies between $y=x^2$ and $y=x^3$ ?

Answer 1

Hint

If $0<x<1$ and $0<a<b$ then $0<x^b<x^a<1$ and if $x>1$ and $0<a<b$ then $1<x^a<x^b.$

Answer 2

For each value of $x$, the value of $x^{5/2}$ lies between the values $x^{4/2}$ and $x^{6/2}$. So for each value of x, the point on the graph of $y=x^{5/2}$ lies between the points on the graph of $y=x^{4/2}$ and $y=x^{6/2}$. When we connect all of those infinitely many points together, the resulting line for the graph of $y=x^{5/2}$ lies between the graphs of $y=x^{4/2}$ and $y=x^{6/2}$.

Answer 3

If $x>1$, $$x^2<\sqrt{x}\cdot x^2=x^{5/2}<\sqrt{x}\cdot x^{5/2}=x^3$$ which follows from the fact that $x^{1/2}=\sqrt{x}>1$ and that $x^a\cdot x^b=x^{a+b}$

When $x<1$, $\sqrt{x}<1$ and by a similar argument, the inequality reverses itself to $x^2>x^{5/2}>x^3$