Answer
The friend is wrong because the sum of the exponents is wrong and the product of 4 and 3 is 12, not 7.
The correct answer is 12$a^{\frac{7}{10}}$
Work Step by Step
4$a^{\frac{1}{2}}$ $\times$ 3$a^{\frac{1}{5}}$
= (4$\times$3)($a^{\frac{1}{2}+\frac{1}{5}})$
= 12$a^{\frac{1(5)+1(2)}{10}}$
=12$a^{\frac{7}{10}}$
