Chaining Exponent Rules Together I'm having trouble understanding why the following property is true and want to make sense of it before going ahead and using it in my proof by induction:
$$2^{2^n}=2^{2^{n-1}}\times 2^{2^{n-1}}=\left( 2^{2^{n-1}} \right)^2$$
I thought that the $2$ outside the bracket would multiply into $n-1$ and result in $2^{2^{2n-2}}\ldots$
Why is the result $2^{2^n}$? How does the  $-1$ in $n-1$ get cancelled out?

$$
2^{2^{n-1}} \times 2^{2^{n-1}} = 2^{2^{n-1}+2^{n-1}} = 2^{2 \times 2^{n-1}} = 2^{2^{1+(n-1)}} = 2^{2^{n}}
$$
