Porve that b^(r+s) = b^r * b^s, r and s are rationals.
Let $\displaystyle r=\frac{m}{n}, \ s=\frac{p}{q}, \ m,n,p,q\in\mathbb{N}, \ n,q\neq 0$
Then $\displaystyle b^{r+s}=b^{\displaystyle\frac{m}{n}+\frac{p}{q}}=b ^{\displaystyle\frac{mq+np}{nq}}=$
$\displaystyle =\displaystyle\sqrt[nq]{b^{mq+np}}=\sqrt[nq]{b^{mq}b^{np}}=$
$\displaystyle =\sqrt[nq]{b^{mq}}\cdot\sqrt[nq]{b^{np}}=b^{\frac{mq}{nq}}\cdot b^{\frac{np}{nq}}=b^{\frac{m}{n}}\cdot b^{\frac{p}{q}}=b^r\cdot b^s$
In a similar way you can prove if r or s or both are negative.