(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Prove that b^{r+s}=b^{r}b^{s}if r and s are rational.

2. Relevant equations

So far we know the basic field axioms and a a few other things related to powers.

1.) For every real x>0 and every integer n>0 there is one and only one positive real y such that y^{n}=x

2.) if a and b are positive real numbers and n is a positive integer, then (ab)^{1/n}=a^{1/n}b^{1/n}

3.)(b^{a})^{b}=b^{ab}

3. The attempt at a solution

When I look at this problem I don't see any way to use the three facts above. The first thing that jumps at me is the field axiom of multiplicative associativity. So for me I see the proof as going as such.

Asssume r and s are rational. b^{r+s}=b^{r}b^{s}due to multiplicative associativity. (QED)

Is the proof this simple or am I missing something?

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# Proof of adding powers (real analysis)

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