Question

The number $(\sqrt{2} + \sqrt{3})^2$ is :
(A) a rational number
(B) an irrational number
(C) an integer
(D) a natural number

Answer 1

$(\sqrt{2} + \sqrt{3})^2 \\ \\ = {( \sqrt{2} )}^{2} + {( \sqrt{3} )}^{2} + 2 \times \sqrt{2} \times \sqrt{3} \\ \\ = 2 + 3 + 2 \sqrt{6} \\ \\ = 5 + 2 \sqrt{6}$

(5 + 2√6) is an irrational number.

Answer is B.

Answer 2

Answer:

(B) an irrational number

Step-by-step explanation:

(√2 + √3)²

Apply (a + b)² = a² + 2ab + b²

= (√2)² + 2(√2)(√3) + (√3)²

Evaluate each term:

= 2 + 2(√6) + 3

Combine like term:

= 5 + 2√6

Since √6 is a irrational number

⇒ 5 + 2√6 is irrational

Answer: (B) an irrational number