Question
NOTE: In the bottom right corner of each question, you will see an option to check answer. Click this to check your answer before submitting. It will not tell you the correct answer, but it will tell you if the answer you selected is correct. You can use this feature once per question.
Drag and drop the option that would make each statement true.
(2 points)
Put responses in the correct response input area to answer the question. Select a response, navigate to the desired input area and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The product of 312−−√
and Press enter key to drop response here.Response input area would yield a rational number.
The product of 312−−√
and Press enter key to drop response here.Response input area would yield an irrational number.
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6√2
5√3
Drag and drop the option that would make each statement true.
(2 points)
Put responses in the correct response input area to answer the question. Select a response, navigate to the desired input area and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The product of 312−−√
and Press enter key to drop response here.Response input area would yield a rational number.
The product of 312−−√
and Press enter key to drop response here.Response input area would yield an irrational number.
Skip to navigation
6√2
5√3
Answers
GPT-4o mini
To determine which response would yield a rational or irrational product when multiplied by \( \sqrt{312} \), we need to assess the nature of the numbers involved.
1. **Rational Number**: A rational number is any number that can be expressed as the quotient of two integers. The product of a rational number and a rational number is rational.
2. **Irrational Number**: An irrational number is a number that cannot be expressed as a fraction of two integers. The product of a rational number and an irrational number is irrational.
Now we evaluate the given options:
- **Rational Number**: Neither \( 6\sqrt{2} \) nor \( 5\sqrt{3} \) are rational numbers as both \( \sqrt{2} \) and \( \sqrt{3} \) are irrational.
- **Irrational Number**: Both options \( 6\sqrt{2} \) and \( 5\sqrt{3} \) are irrational numbers.
Now, if we multiply \( \sqrt{312} \) (which is irrational) by 6√2 or 5√3:
- **Multiplying \( \sqrt{312} \) by \( 6\sqrt{2} \)** or **\( 5\sqrt{3} \)** will yield irrational results.
Therefore, we need to fill in the statements appropriately:
1. The product of \( \sqrt{312} \) and **5√3** would yield a rational number. (This is incorrect: it actually yields an irrational number.)
2. The product of \( \sqrt{312} \) and **6√2** would yield an irrational number.
So the correct placement would be:
- For the first statement, you would insert **6√2** (which does not yield a rational number, so this might be misleading).
- For the second statement, insert **5√3**.
### Summary of Answers:
- The product of \( \sqrt{312} \) and **6√2** would yield a rational number.
- The product of \( \sqrt{312} \) and **5√3** would yield an irrational number.
You should verify these selections with the interface available.
1. **Rational Number**: A rational number is any number that can be expressed as the quotient of two integers. The product of a rational number and a rational number is rational.
2. **Irrational Number**: An irrational number is a number that cannot be expressed as a fraction of two integers. The product of a rational number and an irrational number is irrational.
Now we evaluate the given options:
- **Rational Number**: Neither \( 6\sqrt{2} \) nor \( 5\sqrt{3} \) are rational numbers as both \( \sqrt{2} \) and \( \sqrt{3} \) are irrational.
- **Irrational Number**: Both options \( 6\sqrt{2} \) and \( 5\sqrt{3} \) are irrational numbers.
Now, if we multiply \( \sqrt{312} \) (which is irrational) by 6√2 or 5√3:
- **Multiplying \( \sqrt{312} \) by \( 6\sqrt{2} \)** or **\( 5\sqrt{3} \)** will yield irrational results.
Therefore, we need to fill in the statements appropriately:
1. The product of \( \sqrt{312} \) and **5√3** would yield a rational number. (This is incorrect: it actually yields an irrational number.)
2. The product of \( \sqrt{312} \) and **6√2** would yield an irrational number.
So the correct placement would be:
- For the first statement, you would insert **6√2** (which does not yield a rational number, so this might be misleading).
- For the second statement, insert **5√3**.
### Summary of Answers:
- The product of \( \sqrt{312} \) and **6√2** would yield a rational number.
- The product of \( \sqrt{312} \) and **5√3** would yield an irrational number.
You should verify these selections with the interface available.