Question

Using proof by contradiction or otherwise prove (2)¹/² +(3)¹/²is irrational

Answer 1

To prove that √2 + √3 is irrational, one can assume the opposite and show this leads to a contradiction, thus proving the original statement.

Step-by-step explanation:

To prove that √2 + √3 is irrational, we can use proof by contradiction. We start by assuming the opposite, that √2 + √3 is rational. This means that it can be expressed as a fraction of two integers, p/q, where q is not zero and p and q have no common factors. Following the steps:

1. Assume √2 + √3 is rational. So, √2 + √3 = p/q
2. Multiply both sides by q, we get q√2 + q√3 = p.
3. Square both sides to eliminate the square roots, q²*2 + 2*q²*√2*√3 + q²*3 = p².
4. This simplifies to 2q² + 3q² + 2*q²√6 = p².
5. If √6 were rational, 2*q²√6 would be an integer, making p² a sum of three integers, which makes p also an integer. However, we know that √6 is irrational.
6. Since √6 is irrational, 2*q²√6 cannot be an integer, thus p² cannot be expressed as the sum of integers, contradicting the assumption that p is an integer.
7. Therefore, our initial assumption that √2 + √3 is rational must be false.
8. Hence, √2 + √3 is irrational.

Through this contradiction, we have shown that √2 + √3 cannot be rational and is therefore irrational.