What is the tens place digit of (12)^(42)
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
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
To find the tens place digit of (12)^(42), we need to evaluate the value of (12)^(42) first.
To calculate this, we can use the following steps:
1. Start by calculating the ones place digit of (12)^1, (12)^2, (12)^3, and so on, until we find a pattern.
(12)^1 = 12
(12)^2 = 144
(12)^3 = 1728
(12)^4 = 20736
2. Observe the pattern of the ones place digits: 2, 4, 8, 6, 2, 4, 8, 6, and so on. We notice that the ones place digits repeat every 4 powers.
3. Divide the exponent, 42, by 4: 42 รท 4 = 10 with a remainder of 2. This means that (12)^(42) can be expressed as (12)^(4 * 10 + 2).
4. Simplify the expression: (12)^(42) = (12)^(4 * 10 + 2) = ((12)^4)^10 * (12)^2.
5. Evaluate the values of (12)^4 and (12)^2:
(12)^4 = 20736
(12)^2 = 144
6. Substitute these values back into the expression:
(12)^(42) = ((12)^4)^10 * (12)^2 = 20736^10 * 144.
7. Calculate 20736^10 on a calculator or a computer to get the value. The result is a very large number, but we are only interested in the tens place digit.
8. Consider the tens place digit of the result obtained in step 7. This will be the tens place digit of (12)^(42).