To solve the expression \(8 \times 1000 + 4 \times 100 + 7 \times 10 + 1 \times 1 + 7 \times \frac{1}{10} + 1 \times \frac{1}{100} + 6 \times \frac{1}{1000}\), we will calculate each term separately and then sum them up.
- \(8 \times 1000 = 8000\)
- \(4 \times 100 = 400\)
- \(7 \times 10 = 70\)
- \(1 \times 1 = 1\)
- \(7 \times \frac{1}{10} = 0.7\)
- \(1 \times \frac{1}{100} = 0.01\)
- \(6 \times \frac{1}{1000} = 0.006\)
Now, let's add all these results together:
\[ 8000 + 400 + 70 + 1 + 0.7 + 0.01 + 0.006 \]
Adding them step by step:
\[ 8000 + 400 = 8400 \] \[ 8400 + 70 = 8470 \] \[ 8470 + 1 = 8471 \] \[ 8471 + 0.7 = 8471.7 \] \[ 8471.7 + 0.01 = 8471.71 \] \[ 8471.71 + 0.006 = 8471.716 \]
Therefore, the final result is:
\[ \boxed{8471.716} \]