Rewriting the difference of two $3/2$-powers

Question

There's this part in this problem where it goes $\frac{8}{27}\left[\left(\frac{22}{4}\right)^{3/2} - \left(\frac{13}{4}\right)^{3/2}\right]$ and it equals $\frac{22\sqrt{22} - 13\sqrt{13}}{27}$. If anyone could explain this to me?

Answer 1

Note that for $x \ge 0$ we have $x^{3/2} = x^1 \cdot x^{1/2} = x\sqrt{x}$.

Therefore, $\dfrac{8}{27}\left[\left(\dfrac{22}{4}\right)^{3/2}-\left(\dfrac{13}{4}\right)^{3/2}\right] = \dfrac{8}{27}\left[\dfrac{22\sqrt{22}}{4\sqrt{4}}-\dfrac{13\sqrt{13}}{4\sqrt{4}}\right]$.

Can you simplify things from here? Hint: $4\sqrt{4} = 4 \cdot 2 = 8$.