How to solve

$(20$ points$)$ Let be $a>0,$ we define the exponential function $a^x$:$=e^{ln(a)x}.$ Prove using the definition of exponential functions that

$($a$)(3$ points$)a^{b+c}=a^b{a}^c$

$($b$)(3$ points$)(a^b{)}^c=a^{bc}$

$($c$)(3$ points$)a^n=a.a*cdotsa$ for all ninN

$($d$)(3$ points$)a^{\frac{1}{n}}=\sqrt[n]{a}$ for all ninN

$($e$)(3$ points$)a^{\frac{m}{n}}=\sqrt[n]{a^m}$ for all ninN,minZ

$($f$)(5$ points$)a^x$ is strictly increasing for $a>1$ and strictly decreasing for $0.$