Question
The product of a multiplication problem is 225−−−√ . What do you know about the factors?(1 point) Responses At least one factor is irrational because the product is irrational. At least one factor is irrational because the product is irrational. One factor is rational, but the other is irrational. One factor is rational, but the other is irrational. It cannot be predicted based on the information given. It cannot be predicted based on the information given. They are both rational because the product is rational.
Answers
GPT-4o mini
The product you mentioned, \(\sqrt{225}\), simplifies to 15, which is a rational number. According to the properties of rational and irrational numbers:
- The product of two rational numbers is always rational.
- The product of a rational number and an irrational number is always irrational.
- The product of two irrational numbers can be either rational or irrational.
Given that \(\sqrt{225} = 15\) is rational, the factors involved in a multiplication problem that results in a rational product can either both be rational, or one can be irrational and the other rational. However, in this case, since \(\sqrt{225}\) is rational, we can conclude that at least one factor must also be rational, and both factors could be rational.
Therefore, the correct response would be:
**They are both rational because the product is rational.**
- The product of two rational numbers is always rational.
- The product of a rational number and an irrational number is always irrational.
- The product of two irrational numbers can be either rational or irrational.
Given that \(\sqrt{225} = 15\) is rational, the factors involved in a multiplication problem that results in a rational product can either both be rational, or one can be irrational and the other rational. However, in this case, since \(\sqrt{225}\) is rational, we can conclude that at least one factor must also be rational, and both factors could be rational.
Therefore, the correct response would be:
**They are both rational because the product is rational.**