To add the fractions \( \frac{1}{2} \) and \( \frac{5}{7} \), we first need a common denominator. The denominators here are 2 and 7. The least common multiple (LCM) of 2 and 7 is 14.
Next, we'll convert each fraction to have a denominator of 14:
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Convert \( \frac{1}{2} \) to a denominator of 14: \[ \frac{1}{2} = \frac{1 \times 7}{2 \times 7} = \frac{7}{14} \]
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Convert \( \frac{5}{7} \) to a denominator of 14: \[ \frac{5}{7} = \frac{5 \times 2}{7 \times 2} = \frac{10}{14} \]
Now we can add the two fractions: \[ \frac{7}{14} + \frac{10}{14} = \frac{7 + 10}{14} = \frac{17}{14} \]
The result of \( \frac{1}{2} + \frac{5}{7} \) is \( \frac{17}{14} \).
Thus, the final answer is: \[ [17] [14] \]