Exponentation problems

i have a  $${\sqrt{{\mathtt{a}}}}$$  and the same as  $${{\mathtt{a}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right)}$$ ;

then it is equal to  $${{\mathtt{a}}}^{\left({\mathtt{1}}{\mathtt{\,\times\,}}\left({\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right)\right)}$$ ; 

according to the identities ( $${{\left({{\mathtt{a}}}^{{\mathtt{n}}}\right)}}^{{\mathtt{k}}} = {{\mathtt{a}}}^{\left({{\mathtt{n}}}^{{\mathtt{k}}}\right)} = {{\mathtt{a}}}^{\left({\mathtt{n}}{\mathtt{\,\times\,}}{\mathtt{k}}\right)}$$ ) it is the same as  $${{\mathtt{a}}}^{\left({{\mathtt{1}}}^{\left({\frac{{\mathtt{1}}}{{\mathtt{2}}}}\right)}\right)}$$ ;

in this case it is the same as  $${{\mathtt{a}}}^{{\sqrt{{\mathtt{1}}}}} = {{\mathtt{a}}}^{{\mathtt{1}}}$$ ;

$${{\mathtt{a}}}^{{\mathtt{1}}} = {\mathtt{a}}$$ ;

SO: I have proven a false statement  $${\mathtt{a}} = {\sqrt{{\mathtt{a}}}}$$ ;

could you help me understand the problem?