Which product will be rational?
(1 point)
Responses
17 point Modifying Above 12 with bar times 33
Image with alt text: 17 point Modifying Above 12 with bar times 33
start root 3 end root times start root 9 end root
Image with alt text: start root 3 end root times start root 9 end root
4 times pi
Image with alt text: 4 times pi
negative start root 20 end root times 15
The rational product is: 4 times pi
A whole number is added to a number with two digits after the decimal point. To make sure the answer is reasonable, how many digits should the sum have after the decimal point?
(1 point)
Responses
none
none
1
1
2
2
infinitely many
To determine which of these expressions results in a rational product, we need to consider their properties.
1. 17 point Modifying Above 12 with bar times 33: It is unclear what "Modifying Above 12 with bar" refers to, so it is difficult to determine if the product would be rational or not.
2. start root 3 end root times start root 9 end root: The square root of 3 multiplied by the square root of 9 equals the square root of 27. Since the square root of 27 is not a rational number, the product is not rational.
3. 4 times pi: 4 multiplied by pi is not a rational number since pi is an irrational number. Therefore, the product is not rational.
4. negative start root 20 end root times 15: The negative of the square root of 20 multiplied by 15 results in a negative number. However, since the square root of 20 is not a rational number, the product is not rational.
Therefore, none of these expressions result in a rational product.
To determine which product is a rational number, we need to identify whether it can be expressed as a fraction of two integers.
For the given options:
1. 17 point Modifying Above 12 with bar times 33: This expression seems to be a combination of symbols that is not clear. The meaning of "Modifying Above 12 with bar" is undefined. Therefore, we cannot determine if this product is rational or not without further information.
2. √3 * √9: To simplify this product, we can multiply the square roots and simplify the expression.
√3 * √9 = √(3 * 9) = √27 ≈ 5.196
Since √27 cannot be expressed as a fraction of two integers, this product is irrational.
3. 4 * π: This expression represents a product of the number 4 and the constant π (pi). π is an irrational number, but when multiplied by a rational number (4), the resulting product is still irrational.
4. -√20 * 15: Similar to the previous example, let's simplify this product.
-√20 * 15 = -(√20) * 15 = - (√(4 * 5)) * 15 = - (2√5) * 15 = - 30√5
Since -30 is a rational number and √5 cannot be expressed as a fraction of two integers, this product is irrational.
Therefore, out of the given options, the only rational product is 4 * π.