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How do you express  \(2^2\times 4^2\times 8^2\times 16^2\times ...\times 1024^2 \) as a power of 2?

 May 7, 2021
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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

 May 7, 2021

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