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

Question

Responses

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

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

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.

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

Answer 1

The best response to how the Closure Property proves that the sum of a rational number and an irrational number is irrational is:

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

However, this statement is actually incorrect. The correct statement is that the sum of a rational number and an irrational number is an irrational number. The Closure Property specifies that the result of certain operations (like addition, subtraction, etc.) within a set (like rational numbers) results in a value that is also within that same set. Since the sum of a rational and an irrational number does not result in a rational number, it indicates that the sum must be irrational, thus proving the statement.

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

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

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

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

No, some radicals simplify to rational numbers.

Answer 3

The correct response to Mia's statement is:

No, some radicals simplify to rational numbers.

This is accurate because not all radicals are irrational. For example, \(\sqrt{4} = 2\), which is a rational number. Only radicals that do not simplify to rational numbers are irrational. Therefore, Mia's assertion that there is always an irrational number when a radical is involved in an addition expression is not correct.

Answer 4

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

a rational number
a rational number

an integer
an integer

a whole number
a whole number

an irrational number

Answer 5

To determine the result of the expression \( \sqrt{19} - 32 \), we analyze the components:

1. \(\sqrt{19}\): This is an irrational number because 19 is not a perfect square.
2. \(-32\): This is a rational number (specifically, a negative integer).

When you subtract a rational number from an irrational number, the result is always 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

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

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

Answer 7

The best response to how to prove that a rational number multiplied by an irrational number gives an irrational product is:

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

This approach involves demonstrating that if you take a rational number (say \( r \)) and an irrational number (say \( x \)), their product \( r \cdot x \) cannot be rational. For example, you could assume for contradiction that \( r \cdot x \) is rational and show that this would imply that \( x \) can be expressed as a fraction, contradicting the assumption that it is irrational. Thus, you can prove that the product is indeed irrational.

Answer 8

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

4.2974 . . . ; an irrational number
4.2974 . . . ; an irrational number

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

Answer 9

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

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

2. Now, calculate \( 56 \times \sqrt{12} \): \[ 56 \cdot \sqrt{12} = 56 \cdot 2\sqrt{3} = 112\sqrt{3} \]

3. Next, approximate \( \sqrt{3} \): \[ \sqrt{3} \approx 1.732 \] Therefore, \[ 112\sqrt{3} \approx 112 \times 1.732 \approx 193.824 \]

Now, let's determine the type of the result, which is \( 112\sqrt{3} \). Since \( \sqrt{3} \) is irrational, any non-zero rational number multiplied by an irrational number will yield an irrational number.

Thus, the product \( 112\sqrt{3} \) is an irrational number.

However, since the options provided do not match the approximate product explicitly presented, the correct approach would correspond with knowing the output continues to be an irrational number, as stated.

The final answer can be summarized as: 112√3; an irrational number.