Prove that the sum of 3/4 and sqrt 10 is irrational. Show your work, and in 2 sentences, explain why the sum is an irrational number. Make it simple and short.

Answer 1

To prove that \( \frac{3}{4} + \sqrt{10} \) is irrational, we start by noting that \( \frac{3}{4} \) is a rational number (as it can be expressed as a fraction of integers) and \( \sqrt{10} \) is an irrational number (since 10 is not a perfect square).

The sum of a rational number \( r \) and an irrational number \( s \) is always irrational. Therefore, since \( \frac{3}{4} \) is rational and \( \sqrt{10} \) is irrational, their sum \( \frac{3}{4} + \sqrt{10} \) must also be irrational.

Thus, \( \frac{3}{4} + \sqrt{10} \) is irrational.