To find the fraction of the book James read each night, we start by dividing the total fraction of the book he read, which is \( \frac{6}{7} \), by the number of nights he read, which is 8.
The amount he read each night can be calculated as:
\[ \text{Fraction read each night} = \frac{\frac{6}{7}}{8} \]
We can express 8 as a fraction:
\[ \text{Fraction read each night} = \frac{6}{7} \div 8 = \frac{6}{7} \times \frac{1}{8} \]
Now we multiply the fractions:
\[ \text{Fraction read each night} = \frac{6 \times 1}{7 \times 8} = \frac{6}{56} \]
Next, we simplify \( \frac{6}{56} \):
\[ \frac{6}{56} = \frac{3}{28} \]
Thus, the fraction of the book that James read each night is:
\[ \frac{3}{28} \]
Therefore, the answer is:
\[ \boxed{\frac{3}{28}} \]
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