Adding $\left(\frac{b}{c}\right)^{1/2}+\left(\frac{c}{b}\right)^{3/2}$ I am trying to refresh my algebra skills by reading the book Elements Of Algebra by Leonhard Euler. I came across a problem:

$$\text{Add}\;\left(\frac{b}{c}\right)^{1/2}\;\text{to}\;\left(\frac{c}{b}\right)^{3/2}. \qquad\qquad \text{Ans.}\;\frac{b^2+c^2}{b\sqrt{bc}}$$ 

I don't know how he did it. My understanding is that you can't add radicals that are not the same; for example, $\sqrt{5} + \sqrt{3}$
Could someone enlighten me? Or did he use a some properties of radicals?

\begin{align*}
\left(\frac{b}{c}\right)^{1/2} + \left(\frac{c}{b}\right)^{3/2} = \frac{\sqrt{b}}{\sqrt{c}} + \frac{\sqrt{c^{3}}}{\sqrt{b^{3}}} = \frac{\sqrt{b}\times\sqrt{b^{3}} + \sqrt{c}\times\sqrt{c^{3}}}{\sqrt{cb^{3}}} = \frac{b^{2}+c^{2}}{b\sqrt{bc}}
\end{align*}
