Step 1 :First, we need to remember that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
Step 2 :So, we need to find the reciprocal of \( \frac{8}{7} \). The reciprocal of \( \frac{8}{7} \) is \( \frac{7}{8} \).
Step 3 :Next, we multiply \( \frac{1}{7} \) by \( \frac{7}{8} \).
Step 4 :The result of the multiplication is \( \frac{1}{8} \).
Step 5 :Final Answer: The result of the division \( \frac{1}{7} \div \frac{8}{7} \) in lowest terms is \( \boxed{\frac{1}{8}} \).