To prove that the sum \( \frac{3}{4} + \sqrt{10} \) is irrational, we can use the properties of rational and irrational numbers.
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Identify the components: We have:
- \( \frac{3}{4} \) (which is a rational number)
- \( \sqrt{10} \) (which is an irrational number)
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Understanding rational and irrational sums: The sum of a rational number and an irrational number is always irrational. This is because if we assume \( \frac{3}{4} + \sqrt{10} \) is rational, we can set: \[ x = \frac{3}{4} + \sqrt{10} \] Rearranging gives: \[ \sqrt{10} = x - \frac{3}{4} \] Since both \( x \) and \( \frac{3}{4} \) are rational, their difference \( x - \frac{3}{4} \) should also be rational. However, this would imply \( \sqrt{10} \) is rational, which contradicts the fact that \( \sqrt{10} \) is irrational.
Therefore, the sum \( \frac{3}{4} + \sqrt{10} \) is irrational.