Roots (Integers)

This section is for integers only because there are a lot of shortcuts on how to get roots of perfect powers.

Square roots

So we look at all the digits until the 3rd last digit and then the last 2, and then most of the time we have 2 possibilities. (Sometimes we only have 1, and that is if the number/answer ends in 0/5.

Example: sqrt(24649)

         Look at 246|49

         -> floor(sqrt(246)) = 15

         Last digit is 9: -> 3 or 7

         So our answers are 153 or 157.

         If you're not desperate you can answer with one then another.

         If you can only submit your answer 1 time, then you'll have to find out the exact answer.

         Back to our example: sqrt(24649) = 153 or 157?

         Take 246 - 15^2 = 21 which is larger than 15, so our answer is 157.

Cube roots

For cube roots, we can immediately tell the answer from the last digit!

Example: cbrt(328509)

         Look at 328|509

         -> floor(cbrt(328)) = 6

         Mostly the last digit is the same with the answer, but there are exceptions:

         2 <-> 8 and 3 <-> 7.

         Well the last digit is 9 so we have nothing to worry about.

         -> cbrt(328509) = 69.

Fourth roots

So this is even worse than square roots!

End digits have only 4 possibilities: 0, 1, 5 and 6.

0 -> 0, 

1 -> 1, 3, 7, 9,

5 -> 5, and

6 -> 2, 4, 6, 8.

I don't really have any way of getting the exact value yet, but I'll try and find out how to do these.

More coming soon...