### Instruction: Find the $1314^{\text{th}}$ digit past the decimal point in the decimal expansion of $\dfrac{5}{14}$.### Response: To find the decimal expansion of $\dfrac{5}{14}$, I can divide 5 by 14 using long division.
I know that 14 goes into 50 three times, so the first digit after the decimal point is 3.
Then I have a remainder of 8, so I bring down a zero and divide 80 by 14.
I know that 14 goes into 80 five times, so the next digit is 5.
Then I have a remainder of 10, so I bring down another zero and divide 100 by 14.
I know that 14 goes into 100 seven times, so the next digit is 7.
Then I have a remainder of 2, so I bring down another zero and divide 20 by 14.
I know that 14 goes into 20 once, so the next digit is 1.
Then I have a remainder of 6, so I bring down another zero and divide 60 by 14.
I know that 14 goes into 60 four times, so the next digit is 4.
Then I have a remainder of 4, so I bring down another zero and divide 40 by 14.
I know that 14 goes into 40 two times, so the next digit is 2.
Then I have a remainder of 12, so I bring down another zero and divide 120 by 14.
I know that 14 goes into 120 eight times, so the next digit is 8.
Then I have a remainder of 8, which is the same as before.
This means that the decimal expansion of $\dfrac{5}{14}$ is repeating, and the cycle of digits is 3571428.
To find the $1314^{\text{th}}$ digit past the decimal point, I need to figure out which digit in the cycle corresponds to that position.
I can do this by dividing 1314 by the length of the cycle, which is 6, and looking at the remainder.
If I divide 1314 by 6, I get a quotient of 219 and a remainder of 0.
This means that the $1314^{\text{th}}$ digit is the last digit in the cycle, which is 8.