Hi Jeff. For this, I would use a "guess and check" method.

First I look at the last digit: a 3.

What single-digit number cubed ends in a 3? Only a 7. So the cube root, if it is an integer, ends in a 7.

Now how big is it? 50³ = 125 000 which is a bit too big.
40³ is 64 000, which is too small.

So my guess is 47, which lies between 40 and 50 and ends in a 7:

47 x 47 x 47 = 103823, which is a tiny bit too large ! So the cube root is not an integer.

In fact, this means that the cube root of 103 163 is irrational and hence you can't find an exact decimal value but you can improve the approximation if you wish. We saw that 47³ is slightly too large so try 46.9.

46.9³ = 103161.71 so 46.9 is too small and hence the cube root of 103163 is between 46.9 and 47. If you want more accuracy try something between 46.9 and 47.

Cheers,
Stephen La Rocque and Harley Weston