To evaluate the expression \(8 \times 10^3 + 4 \times 10^2 + 7 \times 10^1 + 1 \times 10^0 + 7 \times \frac{1}{10}^1 + 1 \times \frac{1}{10}^2 + 6 \times \frac{1}{10}^3\), we'll calculate each term step-by-step.
-
Calculate \(8 \times 10^3\): \[ 8 \times 10^3 = 8000 \]
-
Calculate \(4 \times 10^2\): \[ 4 \times 10^2 = 400 \]
-
Calculate \(7 \times 10^1\): \[ 7 \times 10^1 = 70 \]
-
Calculate \(1 \times 10^0\): \[ 1 \times 10^0 = 1 \]
-
Calculate \(7 \times \frac{1}{10}^1\): \[ 7 \times \frac{1}{10}^1 = 7 \times 0.1 = 0.7 \]
-
Calculate \(1 \times \frac{1}{10}^2\): \[ 1 \times \frac{1}{10}^2 = 1 \times 0.01 = 0.01 \]
-
Calculate \(6 \times \frac{1}{10}^3\): \[ 6 \times \frac{1}{10}^3 = 6 \times 0.001 = 0.006 \]
Now, we can add all the calculated values together:
\[ 8000 + 400 + 70 + 1 + 0.7 + 0.01 + 0.006 \]
Calculating step-by-step:
-
First, add the whole numbers: \[ 8000 + 400 + 70 + 1 = 8447 \]
-
Now, add the decimal parts: \[ 8447 + 0.7 = 8447.7 \]
\[ 8447.7 + 0.01 = 8447.71 \]
\[ 8447.71 + 0.006 = 8447.716 \]
Thus, the final result is:
\[ \boxed{8447.716} \]