To prove that the sum \( \frac{3}{4} + \sqrt{10} \) is irrational, we begin by noting that \( \frac{3}{4} \) is a rational number. A rational number can be expressed as the quotient of two integers, specifically \( \frac{3}{4} = \frac{3}{4} \), where both 3 and 4 are integers.
Next, we know that \( \sqrt{10} \) is an irrational number. To prove this, we can show that 10 is not a perfect square; thus, \( \sqrt{10} \) cannot be expressed as a fraction of integers.
Now we consider the sum \( x = \frac{3}{4} + \sqrt{10} \). Assume for contradiction that \( x \) is rational. This would imply that \( \sqrt{10} = x - \frac{3}{4} \) is also rational since the difference of two rationals (in this case, \( x \) and \( \frac{3}{4} \)) is rational. However, this contradicts the established fact that \( \sqrt{10} \) is irrational.
Thus, our assumption must be false, and therefore, the sum \( \frac{3}{4} + \sqrt{10} \) is irrational.
In summary, the number \( \frac{3}{4} \) is rational, and \( \sqrt{10} \) is irrational. The sum of a rational number and an irrational number is always irrational, which confirms that \( \frac{3}{4} + \sqrt{10} \) is indeed irrational.
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