To prove that the expression √5 + √3 + √2 is irrational, we can use a proof by contradiction. Let’s assume that √5 + √3 + √2 is rational. If it is rational, then it can be expressed as a fraction of two integers:
√5 + √3 + √2 = a/b, where a and b are integers, and b ≠ 0.
Next, let’s isolate √5:
√5 = (a/b) – √3 – √2
Now we will square both sides to eliminate the square root:
5 = ((a/b) – √3 – √2)²
Expanding the right side gives us:
5 = (a²/b²) – 2(a/b)(√3 + √2) + (√3 + √2)²
The square of the sum of two square roots will also involve square roots, which complicates matters further. However, we note something critical here: both √3 and √2 are also irrational numbers.
When we add any two irrational numbers, say √3 and √2, the sum may be rational or irrational, but we’ve established that √5 alone cannot yield a rational sum with these irrational values.
Since √5 is irrational, and the square roots of 3 and 2 are also irrational, their combination cannot balance out to produce a rational number.
Conclusively, the assumption that √5 + √3 + √2 is rational leads to a contradiction. Therefore, we can conclude that √5 + √3 + √2 is indeed irrational.