To prove that \( \frac{3}{4} + \sqrt{10} \) is irrational, we can use the property that the sum of a rational number and an irrational number is always irrational.
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Identify the components:
- \( \frac{3}{4} \) is a rational number.
- \( \sqrt{10} \) is an irrational number (since 10 is not a perfect square).
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Sum of a rational number and an irrational number: \[ \text{If } r \text{ is rational and } x \text{ is irrational, then } r + x \text{ is irrational.} \]
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Apply the property:
- Here, \( r = \frac{3}{4} \) (rational) and \( x = \sqrt{10} \) (irrational).
- Therefore, \( \frac{3}{4} + \sqrt{10} \) must be irrational.
Thus, we conclude that the sum \( \frac{3}{4} + \sqrt{10} \) is irrational because it is the sum of a rational number and an irrational number.