How can I prove the following:
Prop.: let be $m,n,p \in \Bbb{N}$ and $a \in \Bbb{R}$ then $$a^m\cdot a^n \cdot a^p=a^{m+n+p}$$ ??? I thinked by induction and I must prove:
1) $a^0\cdot a^0 \cdot a^0=a^{0+0+0}$ is true
2) $a^m\cdot a^n \cdot a^p=a^{m+n+p} \Rightarrow a^{m+1}\cdot a^n \cdot a^p=a^{m+1n+p}$ is true
3) $a^m\cdot a^n \cdot a^p=a^{m+n+p} \Rightarrow a^{m}\cdot a^{n+1} \cdot a^p=a^{m+n+1+p}$ is true
4) $a^m\cdot a^n \cdot a^p=a^{m+n+p} \Rightarrow a^{m}\cdot a^{n} \cdot a^{p+1}=a^{m+n+p+1}$ is true
If 1),2),3),4) are true then Prop. is true
Is it correct?