To find the last digit of the number expressed by [tex]\(4^{2019} + 7^{2020} + 8^{2021} - 9^{2022}\)[/tex], we need to consider the last digit of each term separately and then compute the final result.
1. Finding the last digit of [tex]\(4^{2019}\)[/tex]:
- The last digit of powers of 4 cycle every 2 terms: [tex]\(4, 6\)[/tex].
- Since [tex]\(2019\)[/tex] is odd, the last digit of [tex]\(4^{2019}\)[/tex] is [tex]\(4\)[/tex].
2. Finding the last digit of [tex]\(7^{2020}\)[/tex]:
- The last digit of powers of 7 cycle every 4 terms: [tex]\(7, 9, 3, 1\)[/tex].
- [tex]\(2020 \mod 4 = 0\)[/tex], so the last digit of [tex]\(7^{2020}\)[/tex] corresponds to the 4th term in the cycle, which is [tex]\(1\)[/tex].
3. Finding the last digit of [tex]\(8^{2021}\)[/tex]:
- The last digit of powers of 8 cycle every 4 terms: [tex]\(8, 4, 2, 6\)[/tex].
- [tex]\(2021 \mod 4 = 1\)[/tex], so the last digit of [tex]\(8^{2021}\)[/tex] corresponds to the 1st term in the cycle, which is [tex]\(8\)[/tex].
4. Finding the last digit of [tex]\(9^{2022}\)[/tex]:
- The last digit of powers of 9 cycle every 2 terms: [tex]\(9, 1\)[/tex].
- Since [tex]\(2022\)[/tex] is even, the last digit of [tex]\(9^{2022}\)[/tex] is [tex]\(1\)[/tex].
Now we sum and subtract these last digits to find the last digit of the whole expression:
[tex]\[
4 + 1 + 8 - 1 = 12
\][/tex]
The last digit of [tex]\(12\)[/tex] is [tex]\(2\)[/tex].
Thus, the last digit of [tex]\(4^{2019} + 7^{2020} + 8^{2021} - 9^{2022}\)[/tex] is [tex]\(\boxed{2}\)[/tex].
