Given that $b_1+b_2+\dots+b_n = 1$, how do I find the minimum value of $$\frac{x_1+x_2+\dots+x_n}{x_1^{b_1}x_2^{b_2}\dots x_n^{b_n}}?$$
For $n=2$ I used calculus and found the answer to be $$\frac{1}{b_1^{b_1}b_2^{b_2}}.$$ Extending the concept to higher values of $n$, the desired answer may be guessed as $$\prod\limits_{i=1}^n \left(\frac{1}{b_i}\right)^{b_i}.$$
Is there a better approach?