Question

In your own words in 4 sentences Prove that the sum of 3/4 and square root 10 is irrational. Show your work and then in 1 sentence explain why the sum is an irrational Number

Answer 1

To demonstrate that the sum of \( \frac{3}{4} \) and \( \sqrt{10} \) is irrational, we start by recognizing that \( \frac{3}{4} \) is a rational number (it can be expressed as a fraction of integers). On the other hand, \( \sqrt{10} \) is an irrational number, which can be proven by showing that it cannot be expressed as a fraction of two integers, as its decimal representation is non-repeating and non-terminating. The sum of a rational number and an irrational number is always irrational, meaning that \( \frac{3}{4} + \sqrt{10} \) must also be irrational. Thus, we conclude that \( \frac{3}{4} + \sqrt{10} \) cannot be expressed as a fraction, confirming its irrationality.

In summary, adding a rational number to an irrational number always results in an irrational number, hence \( \frac{3}{4} + \sqrt{10} \) is irrational.