To determine whether each sum is rational or irrational, we can analyze the components of each sum.
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\( \frac{1}{7} + \frac{3}{5} \):
- Both \( \frac{1}{7} \) and \( \frac{3}{5} \) are rational numbers (since both can be expressed as fractions of integers).
- The sum of two rational numbers is also a rational number.
- Therefore, \( \frac{1}{7} + \frac{3}{5} \) is rational.
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\( \pi + 9 \):
- \( \pi \) is an irrational number (it cannot be expressed as a fraction of integers).
- The number 9 is a rational number.
- The sum of a rational number and an irrational number is always irrational.
- Therefore, \( \pi + 9 \) is irrational.
In summary:
- \( \frac{1}{7} + \frac{3}{5} \) is rational.
- \( \pi + 9 \) is irrational.