Question

The product of $\sqrt{2}$ and $(2-\sqrt{2})$ will be:
 

• A. An irrational number
• B. A rational number
• C. An integer
• D. None of the above

Hint:

Multiplying a surd (like $\sqrt{2}$) with a linear expression often produces an irrational result unless special cancellation occurs.

Answer 1

The Correct Option is A

Step 1: Multiply the given terms
\[ \sqrt{2} \times (2-\sqrt{2}) = \sqrt{2} \times 2 - \sqrt{2} \times \sqrt{2} \] \[ = 2\sqrt{2} - 2 \]

Step 2: Classify the result
The expression $2\sqrt{2} - 2$ is the difference of an irrational number ($2\sqrt{2}$) and a rational number ($2$).
Thus, the result is irrational.
\[ \boxed{2\sqrt{2} - 2 \ \text{is irrational}} \]