The product of a multiplication problem is √ 225. What do you know about the factors?

Question

1. They are both rational because product is rational.
2. One factor is rational and the other irrational.
3. At least one factor is irrational because the product is irrational.
4. It cannot be predicted.

Answer 1

To analyze the statement that the product of a multiplication problem is \( \sqrt{225} \), we start by calculating \( \sqrt{225} \):

\[ \sqrt{225} = 15 \]

This value, 15, is a rational number.

In multiplication, the product of two factors can be rational or irrational based on the nature of the factors. Let's examine the options given:

1. They are both rational because the product is rational.
   This is indeed a possibility. If both factors are rational numbers, the product will also be rational. This choice can be true.

2. One factor is rational and the other irrational.
   This is also possible. For example, \( \sqrt{225} = 15 \) could be expressed as \( 5 \times 3 \) (both factors rational) or as \( \sqrt{9} \times \sqrt{25} \) (also rational). You could also express it as \( 15 \times 1 \) (both rational), or some familiar irrational times a rational could yield 15, yet still not exceed rationality. But in the general multiplication sense, this is not a necessity.

3. At least one factor is irrational because the product is irrational.
   This is incorrect, since we have established that \( \sqrt{225} = 15 \) is rational.

4. It cannot be predicted.
   Not true here, since we know the product is rational.

Since we have a rational product (15), the most valid conclusion is:

1. They are both rational because the product is rational.

However, note that while option 2 could be true in another multiplicative scenario, option 1 directly concludes from the evidence presented. Factors may lead to a mixture of rational and irrational possibilities, but under a product of \( \sqrt{225} \), they comfortably lead back to being rational factors predominantly. So the best answer remains option 1.