How did he get the fraction with fraction power?

Question

So we have a simple equation that is from Kepler.

$$\left(\frac{\bar r_1}{\bar r_2}\right)^3 = \left(\frac{T_1}{T_2}\right)^2$$

In an explanation of a physics book, you can resolve for $r_2$ like this:

$$r_2 = r_1 \left(\frac{T_1}{T_2}\right)^{2/3}$$

And I found

$$r_1 = \sqrt[3]{\frac{T_1^2}{T_2^2} r_2^3}$$

First question, is my approach correct? My second and main question is how did he get the $r_2$ equation that I stated first. The physics book doesn't explain how to get from the main equation to $r_2 = r_1 \left(\frac{T_1}{T_2}\right)^{2/3}$. Can someone explain me, please? (By the way, of course the equation for $r_1$ and $r_2$ should be different).

Thank you!

Answer 1

Your equation looks good. Check this out:

$$r_1 = \sqrt[3]{\frac{T_1^2}{T_2^2} r_2^3}= \sqrt[3]{r_2^3}\sqrt[3]{\left(\frac{T_1}{T_2}\right)^2} = r_2\sqrt[3]{\left(\frac{T_1}{T_2}\right)^2}$$

Now all you need to know is that $x^\frac{a}{b}$ is defined as $\sqrt[b]{x^a}$.

Does that help?