To prove that the sum \( \frac{3}{4} + \sqrt{10} \) is irrational, we start by establishing the properties of the numbers involved:
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Identify the nature of the numbers:
- \( \frac{3}{4} \) is a rational number because it can be expressed as the fraction of two integers (3 and 4).
- \( \sqrt{10} \) is an irrational number because 10 is not a perfect square, meaning it cannot be expressed as a fraction of two integers.
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Sum of a rational and an irrational number:
- The sum of a rational number and an irrational number is always irrational. This is a fundamental property of rational and irrational numbers.
- If we assume that \( \frac{3}{4} + \sqrt{10} \) were rational, we could rearrange it to isolate \( \sqrt{10} \): \[ \sqrt{10} = \left( \frac{3}{4} + \sqrt{10} \right) - \frac{3}{4} \] Since \( \frac{3}{4} \) is rational, \( \left( \frac{3}{4} + \sqrt{10} \right) \) being rational means \( \sqrt{10} \) would have to be rational as well, which contradicts the fact that \( \sqrt{10} \) is irrational.
Therefore, the sum \( \frac{3}{4} + \sqrt{10} \) is irrational.
In summary, because the sum of a rational number (here, \( \frac{3}{4} \)) and an irrational number (\( \sqrt{10} \)) is always irrational, we have shown that \( \frac{3}{4} + \sqrt{10} \) is indeed irrational.
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