Final answer:
The number you can multiply by -√2 to get a rational number is -√2 itself; when multiplied together, they result in a rational number, which is 2.
Step-by-step explanation:
To get a rational number by multiplying it with √2, you need to multiply by a number that will eliminate the irrational part of √2. The question is, what number can you multiply by -√2 to get a rational number? The options are a) -√2, b) 2, c) 0, and d) 1/2. To arrive at a rational number, the result of the multiplication must not include any square roots or irrational components.
Multiplying by -√2 (option a) would give you 2, which is indeed a rational number, because (-√2) * (-√2) = √2 * √2 = 2. The product of two negative square roots of the same number results in a positive rational number since the negative signs cancel each other out and the square roots simplify to the number itself. Recall that when two negative numbers multiply, the answer has a positive sign.
Thus, the correct answer to the question is a) -√2.