Final answer:
The units digit of 4 raised to 200 is 6, determined by observing the repeating pattern of the units digits in powers of 4, which cycle every two powers.
Step-by-step explanation:
To determine the units digit (ones digit) of 4 raised to 200, we need to look at the pattern of the units digit of powers of 4. The powers of 4 cycle through the units digits in a pattern: 4, 6, 4, 6, and so on. This is because:
- 4^1 has a units digit of 4.
- 4^2 = 16, which has a units digit of 6.
- 4^3 = 64, which again has a units digit of 4.
- 4^4 = 256, which has a units digit of 6, and the pattern repeats.
Since the pattern repeats every 2 powers, we can determine the units digit for 4^200 by seeing how many times 2 fits into 200. Now, 200 is divisible by 2 exactly 100 times, which means the units digit will match that of 4^2, because we have completed a whole number of cycles.
Therefore, the units digit of 4^200 is 6.