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.
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.
They are both rational because the product is rational.
One factor is rational, but the other is irrational.
. 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.
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.
They are both rational because the product is rational.
One factor is rational, but the other is irrational.
Answers
GPT-4o mini
The correct response in this situation is:
**At least one factor is irrational because the product is irrational.**
This is true because the product \( \sqrt{225} \) simplifies to \( 15 \), which is a rational number. However, if we consider the general case where the product is an irrational number (like \( \sqrt{2} \) for instance), then at least one of the factors must be irrational in such cases. However, in the case of \( \sqrt{225} \), both factors can be rational.
But since the question might refer to another context of \( \sqrt{n} \), if \( n \) is not a perfect square, the factors may likely include at least one irrational number. Thus, the statement regarding irrationality holds in the more general abstract sense for non-perfect-square roots.
**At least one factor is irrational because the product is irrational.**
This is true because the product \( \sqrt{225} \) simplifies to \( 15 \), which is a rational number. However, if we consider the general case where the product is an irrational number (like \( \sqrt{2} \) for instance), then at least one of the factors must be irrational in such cases. However, in the case of \( \sqrt{225} \), both factors can be rational.
But since the question might refer to another context of \( \sqrt{n} \), if \( n \) is not a perfect square, the factors may likely include at least one irrational number. Thus, the statement regarding irrationality holds in the more general abstract sense for non-perfect-square roots.
Abby!!
Put the steps to the proof that the sum t=r+s
is irrational if r is a rational and s is rational in the correct sequence.
Step 1. Subtract, writing s as a fraction.
Step 2. For a contradiction, assume that t is rational, and write r and t as fractions.
Step 3. Realize a contradiction.
Step 4. Find a common denominator.
(1 point)
Responses
Step 3, Step 4, Step 2, Step 1
Step 3, Step 4, Step 2, Step 1
Step 4, Step 2, Step 1, Step 3
Step 4, Step 2, Step 1, Step 3
Step 2, Step 4, Step 1, Step 3
Step 2, Step 4, Step 1, Step 3
Step 1, Step 3, Step 2, Step 4
is irrational if r is a rational and s is rational in the correct sequence.
Step 1. Subtract, writing s as a fraction.
Step 2. For a contradiction, assume that t is rational, and write r and t as fractions.
Step 3. Realize a contradiction.
Step 4. Find a common denominator.
(1 point)
Responses
Step 3, Step 4, Step 2, Step 1
Step 3, Step 4, Step 2, Step 1
Step 4, Step 2, Step 1, Step 3
Step 4, Step 2, Step 1, Step 3
Step 2, Step 4, Step 1, Step 3
Step 2, Step 4, Step 1, Step 3
Step 1, Step 3, Step 2, Step 4
GPT-4o mini
To prove that the sum \( t = r + s \) is irrational if both \( r \) and \( s \) are rational, the correct sequence of steps is:
**Step 2. For a contradiction, assume that \( t \) is rational, and write \( r \) and \( t \) as fractions.**
**Step 1. Subtract, writing s as a fraction.**
**Step 4. Find a common denominator.**
**Step 3. Realize a contradiction.**
Thus, the correct sequence is:
**Step 2, Step 1, Step 4, Step 3.**
However, that exact order doesn't appear in your options. Given the available choices, the closest option would be:
**Step 2, Step 4, Step 1, Step 3.**
If you are mixing up \( s \) with the common denominator step before arriving at a contradiction, it is indeed better represented in that sequence.
**Step 2. For a contradiction, assume that \( t \) is rational, and write \( r \) and \( t \) as fractions.**
**Step 1. Subtract, writing s as a fraction.**
**Step 4. Find a common denominator.**
**Step 3. Realize a contradiction.**
Thus, the correct sequence is:
**Step 2, Step 1, Step 4, Step 3.**
However, that exact order doesn't appear in your options. Given the available choices, the closest option would be:
**Step 2, Step 4, Step 1, Step 3.**
If you are mixing up \( s \) with the common denominator step before arriving at a contradiction, it is indeed better represented in that sequence.