Question
The product of a multiplication problem is √225. What do you know about the factors?
a - One factor is rational, but the other is irrational.
b - At least one factor is irrational because the product is irrational.
c - It cannot be predicted based on the information given.
d - They are both rational because the product is rational.
a - One factor is rational, but the other is irrational.
b - At least one factor is irrational because the product is irrational.
c - It cannot be predicted based on the information given.
d - They are both rational because the product is rational.
Answers
GPT-4o mini
To determine the nature of the factors based on the product being \( \sqrt{225} \):
1. Calculate \( \sqrt{225} \):
\[
\sqrt{225} = 15
\]
Since \( 15 \) is a rational number, we can infer properties about the factors of the multiplication problem that results in \( 15 \).
2. The factors of a product can be rational or irrational. However, in the case of rational products, it's a mathematical principle that if the product of two numbers is rational, then both numbers can be rational or one can be rational while the other is irrational—though if one is irrational, the rational and irrational conditions change depending on the other number.
Given that the product \( 15 \) is rational, the possible outcomes for the factors can be summarized as follows:
- Both factors can be rational (which is common).
- One factor can be rational and the other can be irrational.
- It is impossible for both factors to be irrational since their product must result in a rational number.
Based on this reasoning, we find that, while we have different combinations in terms of rational and irrational factors, we can confirm that if the product is rational (15), at least one of the factors must be rational.
Therefore, the correct answer is:
**d - They are both rational because the product is rational.**
1. Calculate \( \sqrt{225} \):
\[
\sqrt{225} = 15
\]
Since \( 15 \) is a rational number, we can infer properties about the factors of the multiplication problem that results in \( 15 \).
2. The factors of a product can be rational or irrational. However, in the case of rational products, it's a mathematical principle that if the product of two numbers is rational, then both numbers can be rational or one can be rational while the other is irrational—though if one is irrational, the rational and irrational conditions change depending on the other number.
Given that the product \( 15 \) is rational, the possible outcomes for the factors can be summarized as follows:
- Both factors can be rational (which is common).
- One factor can be rational and the other can be irrational.
- It is impossible for both factors to be irrational since their product must result in a rational number.
Based on this reasoning, we find that, while we have different combinations in terms of rational and irrational factors, we can confirm that if the product is rational (15), at least one of the factors must be rational.
Therefore, the correct answer is:
**d - They are both rational because the product is rational.**