How do I solve 8750^1596 on a calculator without getting an overflow error message
Your calculation would result in a value of: 2.785175868 * 10^(6291)
Most calculator would probably only reach a max exponent of 10^100, certainly nowhere near 10^(6291)
Your calculator would always return an error.
One way of doing it would be to do it in stages.
The max exponent on your calculator is probably going to be 100 (i.e. 99)
By experiment, we find that 8750^25 = 3.549779 * 10^98
Now 63*25 = 1575
So, (8750^25)^63 = 8750^(25*63) = 8750^(1575)
Then 8750^(1575) = (3.549779 * 10^98)^63
8750^(1575) = (3.549779^63) * 10^(98*63)
8750^(1575) = (4.599206*10^34) * 10^(98*63)
8750^(1575) = (4.599206*10^34) * 10^(6174)
8750^(1575) = 4.599206*10^(6208)
Also,
8750^21 = 6.055766 * 10^82
Then,
8750^(1596) = 8750^(1575) * 8750^21
8750^(1596) = 4.599206*10^(6208) * 6.055766 * 10^82
8750^(1596) = 4.599206*6.055766 * 10^(6208) * 10^82
8750^(1596) = 27.851719 * 10^(6290)
8750^(1596) = 2.7851719 * 10^(6291)
This is accurate to 4 dp with the first calculated value (using Maple) giving
8750^(1596) = 2.7852 * 10^(6291) to 4 dp