I know you don't mean this, but what I highlighted looks like \(\displaystyle 2^{2}+3\). You obviously mean \(\displaystyle 2^{2+3}\). To indicate that without LaTeX, use grouping symbols:
2^(2+3)

And, of course, 2^2= 4 while 2^3= 8 so (2^2)(2^3)= (4)(8)= 32. More generally, yes, (a^m)(a^n)= a^(m+ n). That's one of the "laws of exponents". The other is (a^m)^n= a^(mn).

please write your question using latex or using parenthesis for better understanding.
(2^2)*(2^3)
here two exponents with same base are multiplied and when the base is same,we add the powers.
So (2^2)*(2^3)=2^(2+3)=2^5=32.
So your answer is correct.