Machir, Sofia, and Lexi are discussing the result of the sum of 8–√+32

Question

Machir, Sofia, and Lexi are discussing the result of the sum of  8–√+32
 .

Machir says the answer is irrational because both numbers are irrational.

Sofia says the answer is rational because both numbers are rational.

Lexi says the result is irrational because one number is irrational and the other is rational.

Who is correct?

1. Machir

2. Sofia

3. Lexi

(1 point)
Person #
 is correct.

GPT-4o mini · Answer

To evaluate the expression $8 - \sqrt{3} + 32$, we can simplify it step by step: 1. First, combine the rational numbers: $$ 8 + 32 = 40 $$ 2. Now, include the irrational component: $$ 40 - \sqrt{3} $$ The term $\sqrt{3}$ is irrational, and any rational number (in this case, 40) minus an irrational number ($\sqrt{3}$) results in an irrational number. Now let's analyze who is correct: - Machir claims the answer is irrational because both numbers are irrational. This is incorrect since 40 is rational. - Sofia states the answer is rational because both numbers are rational, which is also incorrect because the final result is irrational. - Lexi claims the result is irrational because one number is irrational and the other is rational. This statement is correct. Thus, the correct answer is: **3. Lexi** is correct.

bluesanta · Answer

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

an irrational number
an irrational number

a rational number
a rational number

an integer
an integer

a whole number

GPT-4o mini · Answer

To determine the type of number that results from the product of $\sqrt{36}$ and $12$, let's first simplify the expression:

Calculate $\sqrt{36}$: \[ \sqrt{36} = 6 \]
Now, multiply by $12$: \[ 6 \times 12 = 72 \]

The result is $72$, which is a rational number.

So, the correct answer is:

Option #1: rational.

bluesanta · Answer

Elaina, Miguel, and Swati are calculating the product of 6–√
and 23
.

Elaina says the answer is irrational because both numbers are irrational.

Miguel says the answer is rational because both numbers are rational.

Swati says the answer is irrational because one number is rational and the other number is irrational.

Who is correct? Enter the number that corresponds to the correct answer.

Elaina
Miguel
Swati
(1 point)
Person #
is correct.

GPT-4o mini · Answer

To evaluate the product of $\sqrt{6}$ and $23$, let's first identify the nature of the numbers involved: - $\sqrt{6}$ is an **irrational** number. - $23$ is a **rational** number. Now, let's calculate the product: $$ \sqrt{6} \times 23 = 23\sqrt{6} $$ A rational number multiplied by an irrational number results in an **irrational number**. Now, let's evaluate the claims of Elaina, Miguel, and Swati: - **Elaina** says the answer is irrational because both numbers are irrational. (Incorrect, because only $\sqrt{6}$ is irrational.) - **Miguel** says the answer is rational because both numbers are rational. (Incorrect, because $\sqrt{6}$ is not rational.) - **Swati** says the answer is irrational because one number is rational and the other number is irrational. (Correct, because it is true that the product of a rational and an irrational number is irrational.) Thus, the correct answer is: **Person 3 (Swati) is correct.**

bluesanta · Answer

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

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 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 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.

GPT-4o mini · Answer

To show how the closure property proves that the sum of a rational number and an irrational number is irrational, we need to focus on the correct response related to the property and definitions involved.

The closure property states that a set of numbers is closed under an operation (like addition or multiplication) if applying that operation to members of the set yields a member of the same set.

In this context:

The sum of two rational numbers is always a rational number.
However, when you add a rational number to an irrational number, the result cannot be a rational number. If it were rational, then one could isolate the irrational number and show it as rational, which contradicts its definition.

Thus, the correct response should imply that a rational number and an irrational number combined yields an irrational number, not indicating that their sum can be rational.

So the most relevant and correct response here would be:

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

This is true and serves to support the understanding that adding a rational number to an irrational number doesn't yield a rational number, concluding that the sum of a rational and irrational number is irrational. All other responses provided do not effectively demonstrate the point being asked about.

bluesanta · Answer

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.

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.
No, some radicals simplify to rational numbers.

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

GPT-4o mini · Answer

Mia's statement that "anytime there is a radical in an addition expression, it automatically has an irrational number as the sum" is not correct.

The correct response is:

No, some radicals simplify to rational numbers.

For example:

$\sqrt{4} + 2 = 2 + 2 = 4$, which is a rational number.
$\sqrt{9} + 1 = 3 + 1 = 4$, which is also a rational number.

Therefore, while many radicals are indeed irrational (like $\sqrt{2}$, $\sqrt{3}$, etc.), not all radicals are irrational, as they can simplify to rational numbers depending on the square root of a perfect square. Thus, Mia's assertion is incorrect.

bluesanta · Answer

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

an irrational number
an irrational number

a rational number
a rational number

an integer
an integer

a whole number

GPT-4o mini · Answer

To determine the type of number that results from the expression $\sqrt{19} - 32$, let's analyze the components:

$\sqrt{19}$ is an irrational number because $19$ is not a perfect square.
$32$ is a rational number, specifically an integer.

Now, let's evaluate the expression:

\[ \sqrt{19} - 32 \]

Since you are subtracting a rational number (32) from an irrational number ($\sqrt{19}$), the result will still be irrational. This is because the result of a rational number subtracted from an irrational number is always irrational.

Therefore, the correct answer is:

an irrational number.