How does the Closure Property prove that the sum of a rational and irrational number is irrational?(1 point)

Question

Responses

The sum or difference of a rational and irrational number is a rational number.
The sum or difference of a rational and irrational number is a rational number.

Irrational numbers can be added or subtracted.
Irrational numbers can be added or subtracted.

The sum or difference of two rational numbers is a rational number.
The sum or difference of two rational numbers is a rational number.

The sum of an irrational and rational number can’t be predicted

Answer 1

The correct response is:

The sum or difference of a rational and irrational number is a rational number.

However, the statement is incorrect. The accurate statement should be that the sum or difference of a rational number and an irrational number is always an irrational number. The closure property, which states that the sum or product of two numbers of a certain set remains within that set, helps us establish that rational numbers do not "close" around irrational numbers. Thus, when we add a rational number to an irrational number, we cannot form a rational number, confirming that their sum is irrational.

Answer 2

Mia says that anytime there is a radical in an addition expression, it automatically has an irrational number as the sum. Is she correct?(1 point)
Responses

Yes, radicals are irrational.
Yes, radicals are irrational.

No, some radicals simplify to rational numbers.
No, some radicals simplify to rational numbers.

Yes, radicals can’t be simplified, so they must be irrational.
Yes, radicals can’t be simplified, so they must be irrational.

No, radicals always simplify to integers, so they are rational.
No, radicals always simplify to integers, so they are rational.
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Answer 3

The correct response is:

No, some radicals simplify to rational numbers.

Mia's statement is not correct because while many radicals (such as √2, √3, etc.) are indeed irrational, some radicals simplify to rational numbers (for example, √4 = 2 or √9 = 3). Therefore, not every radical in an addition expression guarantees that the sum will include an irrational number.

Answer 4

What type of number will result from the expression 19−−√−32?(1 point)
Responses

a whole number
a whole number

a rational number
a rational number

an integer
an integer

an irrational number
an irrational number

Answer 5

The expression \( \sqrt{19} - 32 \) can be analyzed as follows:

1. Evaluate \( \sqrt{19} \): The square root of 19 is an irrational number because it cannot be expressed as a fraction of two integers.
2. Subtract 32: When you subtract a rational number (32) from an irrational number (\( \sqrt{19} \)), the result remains irrational.

Therefore, the correct response is:

an irrational number.

Answer 6

Ryiaz claims it is impossible to prove that a rational number multiplied by an irrational number gives an irrational product. How can you prove this?(1 point)
Responses

He’s correct; it is impossible to prove in any way.
He’s correct; it is impossible to prove in any way.

by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer
by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer

by using a proof by contradiction to show multiplication is a closed operation for rational numbers
by using a proof by contradiction to show multiplication is a closed operation for rational numbers

by choosing random numbers to plug into multiplication problems until he’s convinced

Answer 7

The correct response is:

by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer.

To prove that the product of a non-zero rational number and an irrational number is irrational, you could show that if \( r \) is a non-zero rational number and \( x \) is an irrational number, then \( r \cdot x \) cannot be expressed as a fraction of two integers. For example, if you assume that \( r \cdot x \) is rational, you can derive a contradiction showing that this would mean \( x \) must be rational, which is impossible.

Answer 8

Find the product of 12−−√ and 56. What type of number is it?(1 point)
Responses

2.6307 . . . ; an irrational number
2.6307 . . . ; an irrational number

4.1569 . . . ; an irrational number
4.1569 . . . ; an irrational number

2.8867 . . . ; an irrational number
2.8867 . . . ; an irrational number

4.2974 . . . ; an irrational number

Answer 9

To find the product of \( \sqrt{12} \) and \( 56 \):

1. Calculate \( \sqrt{12} \): \[ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} \]

2. Now multiply by \( 56 \): \[ 56 \cdot \sqrt{12} = 56 \cdot 2\sqrt{3} = 112\sqrt{3} \]

3. Next, we estimate \( \sqrt{3} \) (approximately \( 1.732 \)): \[ 112 \cdot \sqrt{3} \approx 112 \cdot 1.732 \approx 193.824 \]

Since \( \sqrt{3} \) is irrational, the product \( 112\sqrt{3} \) is also irrational.

However, there are no products listed in the responses that match exactly, but the type of number resulting from \( 112\sqrt{3} \) is indeed irrational.

Given that the options do not include the correct number but only indicate irrationality, the conclusion is that:

The result is an irrational number. If you need the decimal approximation for \( 112\sqrt{3} \), it would be approximately \( 193.824 \), but that specific number isn't among the choices you've given.