Final answer:
The last digit of (3⁹⁹⁹⁹) is found by recognizing the repeating pattern of the unit digits in the powers of 3.
The pattern repeats every four powers, and since 9,999 is divisible by 4, the unit digit matches that of 3⁴, which is 1.
Step-by-step explanation:
To find the last digit (unit digit) of (3⁹⁹⁹⁹), we must look for a pattern in the powers of 3.
When we calculate powers of 3, we see that they follow a specific pattern for the unit digits:
3¹ has a unit digit of 3.
3² has a unit digit of 9.
3³ has a unit digit of 7.
3⁴ has a unit digit of 1.
This pattern of unit digits (3, 9, 7, 1) repeats every four powers.
Since 9,999 is divisible by 4, the pattern will complete a whole number of cycles, ending with a power equivalent to 3⁴.
Therefore, the unit digit of (3⁹⁹⁹⁹) is 1.