To decide whether the sum [tex]\(\frac{\sqrt{7}}{7} + \frac{14}{23}\)[/tex] represents a rational number or an irrational number, follow these steps:
1. Analyze the first term [tex]\(\frac{\sqrt{7}}{7}\)[/tex]:
- The numerator, [tex]\(\sqrt{7}\)[/tex], is the square root of 7. Since 7 is not a perfect square, [tex]\(\sqrt{7}\)[/tex] is an irrational number.
- Dividing an irrational number by a non-zero rational number (7 in this case) results in another irrational number. Hence, [tex]\(\frac{\sqrt{7}}{7}\)[/tex] is an irrational number.
2. Analyze the second term [tex]\(\frac{14}{23}\)[/tex]:
- The numerator, 14, and the denominator, 23, are both integers.
- The fraction [tex]\(\frac{14}{23}\)[/tex] represents a ratio of two integers, making it a rational number.
3. Analyze the sum [tex]\(\frac{\sqrt{7}}{7} + \frac{14}{23}\)[/tex]:
- The first term is irrational, and the second term is rational.
- The sum of an irrational number and a rational number is always irrational because there's no way to combine a non-repeating, non-terminating decimal with a repeating or terminating decimal to get a repeating or terminating decimal.
Given these steps:
The sum [tex]\(\frac{\sqrt{7}}{7} + \frac{14}{23}\)[/tex] represents an irrational number. This is because the number [tex]\(\frac{\sqrt{7}}{7}\)[/tex] is irrational and the number [tex]\(\frac{14}{23}\)[/tex] is rational. The sum of an irrational number and a rational number is always irrational.
