You are right! This is a very tough problem and no ordinary calculator can handle such big exponents, because (19^19) - 2 =1,978,419,655,660,313,589,123,977 - which is 25-digit Prime Number !!.
However, if your class takes computer coding, it can be handled by writing a short computer code to do the calculation for you. Out of curiosity, I wrote the following code in Python to give you the answer to your question:
%%time
# Fast python code that first calls pow()
# then applies % operator
import math as m
a =19
b =(19 ** 19) - 2
p =10000000000
# Using direct fast method to compute
# (a ^ b) % p.
d = pow(a, b, p)
print("The Remainder =", f"{d :,d}")
The Remainder = 7,185,569,139 - This is the answer to your question.
Wall time: 981 µs
!
