Properties of exponents when dealing with induction.

Question

This will most likely be a simple question for most of you. While watching my lecture today the white board cut out and the instructor didn't explain the final step in an example.

He went from $(3^{2^{i-1}})^2$ to $3^{2^i}$ without any explanation. What exponent property is that?

Answer 1

The general property is $(x^m)^n = x^{mn}$.

In this case,

$$\left(3^{2^{i-1}}\right)^2 = 3^{2^{i-1}\cdot 2} = 3^{2^{(i-1)+1}} = 3^{2^i}. $$

Answer 2

You should know that $$(a^b)^c=a^{b\cdot c}$$ and $$a^b\cdot a=a^{b+1} $$