Proof that (√2 + √3)² is Irrational

We are given that √6 is irrational. We need to prove that:

(√2 + √3)²

Step 1: Expand the Expression

Using the identity (a + b)² = a² + b² + 2ab, we get:

(√2 + √3)² = (√2)² + (√3)² + 2√2√3

= 2 + 3 + 2√6

= 5 + 2√6

Step 2: Analyze the Result

• 5 is a rational number.
• √6 is irrational (given).
• 2√6 is also irrational (product of a non-zero rational and an irrational number).

Step 3: Key Property

Property: The sum of a rational number and an irrational number is always irrational.

Since:

5 (rational) + 2√6 (irrational)

Therefore, their sum is irrational.

Conclusion

(√2 + √3)² = 5 + 2√6 is irrational.

Hence proved.