how do i find the base and exponent of 117649?

Take your calculator and start dividing 117649 by prime 
numbers until you find one that gives you a whole number:

Try 117649÷2.  That gives 58824.5, not a whole number.
Try 117649÷3.  That gives 39216.33333, not a whole number.
Try 117649÷5.  That gives 23529.8, not a whole number.
Try 117649÷7.  That gives 16807, which IS a whole number.

So 117649 = 7×16807

Try 16807÷7. That gives 2401, which IS a whole number.
So 117649 = 7×7×2401

Try 2401÷7. That gives 343, which IS a whole number.
So 117649 = 7×7×7×343.

Try 343÷7. That gives 49, which IS a whole number.
So 117649 = 7×7×7×7×49.

And everybody knows that 49 = 7×7, so

117649 = 7×7×7×7×7×7

And that means

117649 = 7×7×7×7×7×7 = 76.

Therefore the base is 7 and the exponent is 6.

Edwin