MHB Squaring and Cubing Equations: Explained!

[paulmdrdo1]

Can you explain the part of the solution where v2/v1 = a2/a1were being squared and cubed repectively?

Thanks!

 

[mrtwhs]

$$(r^3)^2 = (r^2)^3$$

 

[Evgeny.Makarov]

To provide a little more detail: It is known that the volume of a sphere of radius $r$ is $cr^3$ for some $c$ (in fact, $c=\frac43\pi$) and its area is $dr^2$ ($d=4\pi$). Therefore,
\[
\left(\frac{V_2}{V_1}\right)^2=\left(\frac{cr_2^3}{cr_1^2}\right)^2=\left(\frac{r_2}{r_1}\right)^6
\]
One can similarly show that $(A_2/A_1)^3=(r_2/r_1)^6$ regardless of $d$.