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2.21 Proof of power law

(iii) First suppose that $q=1$ and work out successive cases

\exp(1)=e=e^{1};\exp(2)=\exp(1+1)=\exp(1)\exp(1)=ee=e^{2};

\exp(3)=\exp(1+2)=\exp(1)\exp(2)=ee^{2}=e^{3};

\exp(4)=\exp(1+3)=\exp(1)\exp(3)=ee^{3}=e^{4};

and we continue until we reach $\exp(p)=e^{p}.$

Now suppose that $p,q$ are natural integers; then by repeating the previous argument, we can show that

(\exp p/q)^{q}=\exp(p)=e^{p};

now take $q^{th}$ roots to get

\exp(p/q)=e^{p/q}.

For $q<0$ , we can use (ii).