17. Select the answer with a rational product and a correct explanation for why the product is rational.
(1 point
• V25. because the product of a fraction and a square root is always a rational number
5
03
- • . because the product of a fraction and an irrational number is always a rational number
15
O (0.555.. ) ..
+g, because the product of a repeating decimal and a rational number is always a rational
number
• v2 . 32, because the product of two irrational numbers is always a rational number
V25. because the square root of a perfect square is always a rational number.
Show an example of a rational number that is between 10−−√ and 11−−√.(1 point)
Show an example of a rational number that is between 10−−√ and 11−−√.(1 point)
3.4
3.4
3.1
3.1
10/3
16/5
Bot?
The correct answer is: (1 point) V25. because the product of a fraction and a square root is always a rational number.
Explanation: When multiplying a fraction and a square root, you can rationalize the denominator of the fraction by multiplying both the numerator and denominator by the conjugate of the square root. This would result in a rational expression. So, the product of a fraction and a square root is always rational.
To determine which answer has a rational product and a correct explanation, we need to examine each option:
1) V25: This answer suggests that the product of a fraction and a square root is always a rational number. However, this statement is incorrect. In general, the product of a fraction and a square root can be rational, irrational, or even undefined.
2) 5/03: This answer claims that the product of a fraction and an irrational number is always rational. This is also incorrect. The product of a fraction and an irrational number can be irrational or rational, depending on the specific values involved.
3) (0.555...): This answer suggests that the product of a repeating decimal and a rational number is always rational. This statement is correct. When you multiply a repeating decimal (which can be written as a fraction) by a rational number, the product will always be rational.
4) V2 * 32: This answer claims that the product of two irrational numbers is always rational. This statement is incorrect. The product of two irrational numbers is most often irrational, not rational.
Based on the explanations provided, the correct answer is the third option, (0.555...), because the product of a repeating decimal and a rational number is indeed always rational.