Prove that root 5 is irrational: Prove that 2 + 3

Prove that 2 + 3 5 is an irrational number , given that 5 is an irrational number . Prove that √ 5 is an irrational number.Are the square roots of all positive integers irrational ? If not, give an example of the square root of a number that is a rational number. Prove that 3 + 2 √ 5 is irrational . Identify the following as rational or irrational number. Give the decimal representation of rational number: √ 9 2 7 Identify the following as rational or irrational number. Give the decimal representation of rational number: √ 1 0 0 In the following equation, find which ... Answer: To prove that is irrational , let's use proof by contradiction. 1. Assume the opposite: Suppose is rational. This means it can be written as , where and are integers with no common factors (i.e., is in simplest form). 2. Manipulate the equation: If , then by cross-multiplying, we get: \sqrt { 5 } = \frac {q} {p}. 3. Contradiction: However, it is a well-known fact that is irrational (since 5 is not a perfect square). This contradicts the assumption that is rational. 4. Conclusion ... Prove that √ 5 is an irrational number. Hence, show that -3 + 2√ 5 is an irrational number. Let us consider √ 5 be a rational number, then √ 5 = p/q, where ‘p’ and ‘q’ are integers, q 0 and p, q have no common factors (except 1). So, 5 = p 2 / q 2 p 2 = 5q 2 …. (1) As we know, ‘ 5 ’ divides 5q 2, so ‘ 5 ’ divides p 2 as well.