Question

The product of two irrational numbers \( 3 + \sqrt{6} \) and \( 3 - \sqrt{5} \) will be a/an

• A. Rational number
• B. Irrational number
• C. Integer
• D. Natural number

Hint:

The product of two irrational numbers is typically irrational unless the irrational components specifically counteract each other to produce a rational number.

Answer 1

The Correct Option is B

Step 1: Multiply the two expressions using the distributive property (Foil method): \[ (3 + \sqrt{6})(3 - \sqrt{5}) = 9 - 3\sqrt{5} + 3\sqrt{6} - \sqrt{30} \] Step 2: Simplify the expression: \[ 9 - 3\sqrt{5} + 3\sqrt{6} - \sqrt{30} \] As none of the radicals can be simplified further or combine to form rational numbers, the expression remains irrational. Thus, the correct answer is (B) Irrational number.