What is the 2012th digit after the decimal point in the decimal expansion of 8/81?
8/81 = .098765432 ...
The repeating group is 9 digits
2012/9 = 223 remainder 5
So, the 9 digits repeat 223 times making 2007 digits, and the 2012th digit is the 5th digit of the group = 6
That might actually be right..
To find the 2012th digit after the decimal point in the decimal expansion of 8/81, we can use long division to divide 8 by 81.
Here's the step-by-step process:
1. Set up the long division: Write 8 as the dividend and 81 as the divisor.
```
0.0987654320987654320987...
---------------------
81 | 8.0000000000000000000000...
```
2. Divide the first digit of the dividend (8) by the divisor (81). The quotient is 0.
```
0.0
---------------------
81 | 8.0000000000000000000000...
0
```
3. Bring down the next digit (0) of the dividend. Since no more decimal places are available, add a zero to the right of the decimal point.
```
0.0
---------------------
81 | 8.0000000000000000000000...
80
```
4. Divide the new dividend (80) by the divisor (81). The quotient is 0.
```
0.00
---------------------
81 | 8.0000000000000000000000...
80
--
00
```
5. Continue bringing down zeros and performing the division until you reach the desired number of decimal places. In this case, the desired digit is the 2012th digit.
As you can see, the pattern 0987654320987654320987... repeats every 81 digits after the decimal point. Since the pattern repeats, the 2012th digit would be the same as the 2012 modulo 81th digit in the pattern.
To calculate this, find the remainder when 2012 is divided by 81:
```
2012 mod 81 = 71
```
Therefore, the 2012th digit after the decimal point in the decimal expansion of 8/81 is the same as the 71st digit in the pattern, which is 7.