Question

What is the tens place digit of (12)^(42)

Answer 1

look at patterns
12^1 = 12 , ends in 2
12^2 = 144, ends in 4
12^3 = 1728, ends in 8
12^4 = 20736 ends in 6
12^5 = 248832 ends in 2
12^6 = 2985984 ends in 4
12^7 ..... ends in 8
12^8 .... ends in 6
....
they end in:
2, 4, 8, 6, 2, 4, 8 ,6, ....

using odd exponents, the answer ends in either 2 or 8
our exponent of 42 is even, so those endings our out!

notice if the exponent is even AND divides evenly by 4 , the ending is 6
if the exponent is even AND is not divisible by 4, the ending is 4

so 12^42 will end in a 4

Answer 2

12^42 = (2+10)^42 = 2^42 + 42*2^41*10 + higher powers of 10 stuff, which will not affect the last two digits
2^42 + 420*2^41 = ...42944 and ends in 44

check: 12^42 = ....96544