How do you express \(2^2\times 4^2\times 8^2\times 16^2\times ...\times 1024^2 \) as a power of 2?
Your expression looks like a geometric sequence of powers of 2. This is helpful because if we want to express something as a power of two, we need to first get all the base numbers to be 2.
We'll use our first power rule, that \({(2^a)}^b = 2^{a+b}\)
For example, \(8^2 = {2^3}^2 = 2^{3*2} = 2^6\)
We get \(2^2 * 2^4 * 2^ 6 * ... * 2^{20}\)
Now we get to use another power rule, that \(2^a * 2^b = 2^{a+b}\)
\(2^{2+4+6+...+20} = 2^{110}\)
As a quick tip, you can do that addition quickly by remembering that in a linearly increasing sequence, the sum = (first + last)/2 * # of terms
