# Number Properties Puzzle – Challenge 7

A math puzzle for smart people with a brain for math! This challenge is for those who love to be challenged by mathematics and critical thinking problems. Let's challenge your brain!

## 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.