Answer:
7
Explanation:
Let's look at powers of 3 and if a pattern forms for the ones digit.
3^0 = 1
3^1 = 3
3^2 = 9
3^3 = 27
3^4 = 81
3^5 = 243
3^6 = 729
3^7 = 2187
3^8 = 6561
3^9 = 19683
3^10 = 59049
3^11 = 177147
etc.
Using the table above, we see a repeating pattern for the ones digit:
1, 3, 9, 7, 1, 3, 9, 7, 1, 3, 9, 7, etc.
Now let's look at the exponent of 3 compared to the ones digit.
exponent, ones digit
0, 1
1, 3
2, 9
3, 7
4, 1
5, 3
6, 9
7, 7
etc.
Let's write the exponent of 3 as a multiple of 4 plus an integer followed by the ones digit.
4 × 0 + 0, 1
4 × 0 + 1, 3
4 × 0 + 2, 9
4 × 0 + 3, 7
4 × 1 + 0, 1
4 × 1 + 1, 3
4 × 1 + 2, 9
4 × 1 + 3, 7
etc.
When the exponent is a multiple of 4, the ones digit is 1.
When the exponent is 1 added to a multiple of 4, the ones digit is 3.
When the exponent is 2 added to a multiple of 4, the ones digit is 9.
When the exponent is 3 added to a multiple of 4, the ones digit is 7.
Now let's look at out exponent, 2091.
2091/4 = 522.75
2091 = 4 × 522 + 3
2091 is 3 added to a multiple of 4, so the ones digit is 7.