Number of roots of differentiable function I have two related problems that are causing me big trouble:
Let $\;a_1,...,a_n\in\Bbb R\;,\;\;a_i\neq 0\;\;\forall\;i\;$ , and let $\;b_1,...,b_n\in\Bbb R\;,\;\;b_i\neq b_j\;\;\forall\,i\neq j\;$ .
(1) Prove that the equation
$$a_1x^{b_1}+\ldots+a_nx^{b_n}=0$$
has at most $\;n-1\;$ different zeros in $\;(0,\infty)\;$ (Hint: Use induction)
(2) Prove that the equation
$$a_1e^{b_1x}+\ldots+a_ne^{b_nx}=0$$
has at most $\;n-1\;$ different zeros in $\;\Bbb R\;$ .
Now, the following is what I've done so far: for a differentiable function $\;f\;$ ,  I can prove that if $\;f'(x)=0\;$ has $\;n-1\;$ different zeros, then $\;f(x)\;$ has at most $\;n\;$ different zeros, using the mean value theorem, say.
But I can't see how to use induction in this case: if I differentiate in the first problem, I get $$0=f'(x)=a_1b_1x^{b_1-1}+\ldots +a_nb_nx^{b_n-1}=\frac1x\sum_{i=1}^na_ib_ix^{b_i}$$
...and now?! How does induction kick in here? I though perhap something like: if we suppose WLOG $\;b_1<b_2<...<b_n\;$ , then perhaps inducting on $\;\lfloor b_n\rfloor\;$, but it gets messy and blurry if $\;b_n<0\;$...
Any help, hints are very much appreciated.

Hint:  $a_1 x^{b_1} + \ldots + a_n x^{b_n}$ has the same number of roots in $(0,\infty)$ as $a_1 + \ldots + a_n x^{b_n - b_1}$.
