### Brain Teasers

# 2^1100???

Your professor prepared a rigorous 150-question final exam to be completed in under 30 minutes, but unfortunately, he seems to have misplaced it, and can only remember one problem. So, he has decided to waive your final exam, if you can answer the one question he can remember from his original exam.

Which is greater, 2^1100 or 3^700? Approximately how many times greater is it (to the nearest 10)?

Which is greater, 2^1100 or 3^700? Approximately how many times greater is it (to the nearest 10)?

### Hint

The numbers are too big to deal with on a normal graphics calculator scale, think of it in terms of powers of ten.### Answer

3^700 is greater, by approximately 710 times (711.0220569 to be exact).To start, one must realize these numbers are too big to deal with by hand. So, one must use exponents and the "log" function on a calculator.

2^1100=10^a

2^x=10^1

log(10)/log(2)=x

3.321928095=x

1100/x=331.1329952

2^3.321928095=10^1

2^(331.132995*3.321928095)=10^(331.132995*1)

2^1100=10^331.132995

Now that we know what 2^1100 equals, apply the same concept to 3^700.

3^700=10^b

3^y=10^1

log(10)/log(3)=y

2.095903274=y

3^2.095903274=10^1

3^(333.9848783*2.095903274)=10^(333.9848783*1)

3^700=10^333.9848783

So, now we have 2^1100 and 3^700 in terms of "ten to the power of".

2^1100=10^331.1379952

3^700=10^333.9848783

Obviously, 3^700 is greater. To figure out how much greater, requires a little more work, but not much.

First, subtract the exponents.

333.9848783-331.1379952

This equals 2.851883073

Then, take 10^2.851883073

This equals 711.0220569, which is around 710.

Thus, 3^700 is approximately 710 times greater than 2^1100. Congratulations, you passed your professor's exam.

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