Number Properties Puzzle – Challenge 7

Challenge:

If \(a = (5^{11})^{2}\) and \(b = 4^{11}\), how many digits long is ab?

A- 22

B- 23

C- 33

D- 44

E- 121

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\(a = (511)^{2} = 5^{22}\) and \(b = 4^{11}\)
Change the base of number b from 4 to 2: \(b = 2^{22 }\)
Therefore, \(ab = 5^{22} × 2^{22 }= 10^{22}\)
Number \(10^{1}\) has 2 digits and number \(10^{2} (100)\) has 3 digits. Therefore, \(10^{22}\) has 23 digits.

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