prove that the sum 3/4 and sqrt10 is rational. Show your work, and in 1-2 sentences explain why the sum is irrational

Answer 1

To prove that the sum \( \frac{3}{4} + \sqrt{10} \) is irrational, we can use the properties of rational and irrational numbers.

1. Identify the components: We have:

   • \( \frac{3}{4} \) (which is a rational number)
   • \( \sqrt{10} \) (which is an irrational number)

2. 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.