Simple exponent algebra question

Question

Suppose we have

$$10^{-2^{n+1}} \cdot 10^{a2^{n}}$$

How does this simplify down to

$$10^{(a-2)2^n}$$

Can anyone clarify?

Answer 1

So you have exponentials with the same base [10] and you are multiplying them so we sum up the exponents$$-2^{n+1}+a2^n$$ You can write $-2^{n+1}$ as $-2^n \cdot 2$ and also $2^n \cdot -2$ sampling this we have $$2^n(-2)+2^na$$ here we just take out the common factor $2^n$ and we have $$2^n(-2+a) $$ or $2^n(a-2)$ which is what we were looking for then we plug the base and we have $$10^{2^n(a-2)}$$