To prove that the sum \( \frac{3}{4} + \sqrt{10} \) is irrational, we start by establishing that \( \sqrt{10} \) is irrational. This is because \( 10 \) is not a perfect square, and the square root of any non-square integer is irrational.
Now, if we assume that \( \frac{3}{4} + \sqrt{10} \) is rational, we can write it as \( r \), where \( r \) is a rational number. Then, rearranging gives \( \sqrt{10} = r - \frac{3}{4} \). Since \( r \) and \( \frac{3}{4} \) are both rational, their difference \( r - \frac{3}{4} \) must also be rational, implying that \( \sqrt{10} \) is rational, which is a contradiction.
Thus, the sum \( \frac{3}{4} + \sqrt{10} \) must be irrational. In summary, since the difference of a rational number and a supposedly rational square root leads to a contradiction, it confirms that \( \frac{3}{4} + \sqrt{10} \) is indeed irrational.
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