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

Question

GPT-4o mini · Answer

To show that the sum $ \frac{3}{4} + \sqrt{10} $ is irrational, we start by recognizing the nature of each component in the sum. 1. **Identify if individual components are rational or irrational:** - $ \frac{3}{4} $ is a rational number (as it can be expressed as a fraction of two integers). - $ \sqrt{10} $ is an irrational number (as it cannot be expressed as a fraction of two integers). 2. **Consider the sum of a rational and an irrational number:** - The sum of a rational number and an irrational number is always irrational. This is a fundamental property of rational and irrational numbers. Thus, since $ \frac{3}{4} $ is rational and $ \sqrt{10} $ is irrational, we can conclude: $$ \frac{3}{4} + \sqrt{10} \text{ is irrational.} $$ **Explanation:** The sum $ \frac{3}{4} + \sqrt{10} $ is irrational because the sum of a rational number (in this case, $ \frac{3}{4} $) and an irrational number (here, $ \sqrt{10} $) results in an irrational number, as it cannot be expressed as a fraction of two integers.