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

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.

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

Irrational numbers can be added or subtracted.

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

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

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

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

What type of number will result from the expression 19−−√−3/2
?(1 point)
Responses

an irrational number
an irrational number

a rational number
a rational number

a whole number
a whole number

an integer

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

Find the product of 12−−√
and 5/6
. What type of number is it?(1 point)
Responses

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

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

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

4.1569 . . . ; an irrational number

1 answer

Here are the answers to your questions:

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

    • Correct Response: The sum or difference of a rational and irrational number is a rational number. (This statement is incorrect; the correct statement is that the sum or difference of a rational and irrational number is an irrational number. However, given the context, none of the options perfectly reflect this.)

  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?

    • Correct Response: No, some radicals simplify to rational numbers.

  3. What type of number will result from the expression \( \sqrt{19} - \frac{3}{2} \)?

    • Correct Response: an irrational number. (Since \( \sqrt{19} \) is irrational, and subtracting a rational number from it will still give an irrational number.)

  4. 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?

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

  5. Find the product of \( \sqrt{12} \) and \( \frac{5}{6} \). What type of number is it?

    • Correct Response: 2.8867 . . . ; an irrational number. (Since \( \sqrt{12} = 2\sqrt{3} \), and multiplying it by a rational number \( \frac{5}{6} \) results in \( \frac{10\sqrt{3}}{6} \), which is still irrational.)