what is 3^500 in scientific form

Let's examine how we can arrive at  Alan's  answer,  since it might not be immediately intuitive........

Note that  log 3⁵⁰⁰   asks.........What power do we raise 10 to in order to obtain 3⁵⁰⁰  ???

By a property of logs, we can write

log 3⁵⁰⁰    as   500*log3 .........and evaluating this gives us......... 238.56062735983122

Now....note that the "whole number" part of this tells us what power (exponent) will be on the "10" in the scientific notation representation of 3⁵⁰⁰ , and 10 raised to the decimal part will tell us what some of the first few digits of that number will be (within a certain degree of accuracy).

So 10^{.56062735983122} = 3.63602917958700501  (Only the first 12 digits of this are accurate for 3⁵⁰⁰ ... see below)

Thus, in scientific notation we would have......

3.63602917958 X 10²³⁸   

This number is actually 239 digits in length and Alan's answer of   3.636×10²³⁸   is good enough !!!!

However....if you insist.....we have

3636029179586993684238526707954331911802338502600162304034603583258060019158389548419850

8262979388783308179702534403855752855931517013066142992430916562025780021771247847643450

125342836565813209972590371590152578728008385990139795377610001

 I think I'll stick with the scientific notation........

3⁵⁰⁰ ≈ 3.636×10²³⁸