Sums and products of rational number quick check
1. According to the Closure Property, which of these sums results in a rational number?(1 point)
Responses
36−−√+221−−−√
Start Root 36 End Root plus Start Root 221 End Root
0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction
0.6¯¯¯¯+910
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction
π+4–√
pi plus Start Root 4 End Root
2.In the proof that ab+cd, where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbd is a rational number.(1 point)
By the Closure Property, a quotient of imaginary numbers is a rational number.
By the Closure Property, ad+bc and bd are both quotients of integers, and so ad+bcbd is a quotient of two integers.
By the Closure Property, ad+bc and bd are both integers, and so ad+bcbd is a quotient of two integers.
By the Closure Property, ad+bcbd reduces to a whole number, which is a subset of rational number
3.The sum of two rational numbers is always(1 point)
Responses
radical.
zero.
irrational.
rational.
4. The Closure Property implies that the product of 4/5 and 15 is what type of number?(1 point)
irrational
rational
an integer
zero
5. Malanna says that when she multiplies two rational numbers, her product will be irrational. Jolene says that the product in this case will be rational. Who is correct?(1 point)
Responses
Neither of them, it depends on the number.
Malanna
Both options are possible depending on the numbers.
Jolene
How can you rewrite 812 using a root?(1 point)
Responses
1. 12−−√8
root index 8 Start Root Start Fraction 1 over 2 End Fraction End Root
2. 8–√1
root index 1 Start Root 8 End Root
3. 8–√12
root index Start Fraction 1 over 2 End Fraction Start Root 8 End Root
4. 8–√
Start Root 8 End Root
Neither of them, it depends on the numbers.
can i ask you another question
Of course! Feel free to ask another question.
According to the Closure Property, which of these sums results in a rational number?(1 point)
Responses
36−−√+221−−−√
Start Root 36 End Root plus Start Root 221 End Root
0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction
0.6¯¯¯¯+910
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction
π+4–√
According to the Closure Property, the sum that results in a rational number is:
0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction
In the proof that ab+cd, where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbd is a rational number.(1 point)
Responses
By the Closure Property, a quotient of imaginary numbers is a rational number.
By the Closure Property, a quotient of imaginary numbers is a rational number.
By the Closure Property, ad+bc and bd are both quotients of integers, and so ad+bcbd is a quotient of two integers.
By the Closure Property, A d plus b c and b d are both quotients of integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers.
By the Closure Property, ad+bc and bd are both integers, and so ad+bcbd is a quotient of two integers.
By the Closure Property, A d plus b c and b d are both integers, and so the fraction with numerator A d plus b c and denominator b d is a quotient of two integers.
By the Closure Property, ad+bcbd reduces to a whole number, which is a subset of rational numbers.
By the Closure Property, the fraction with numerator A d plus b c and denominator b d reduces to a whole number, which is a subset of rational numbers.
By the Closure Property, ad+bc and bd are both integers, and so ad+bcbd is a quotient of two integers.
The sum of two rational numbers is always(1 point)
Responses
radical.
zero.
irrational.
rational.
The sum of two rational numbers is always rational.
The Closure Property implies that the product of 45 and 15 is what type of number?(1 point)
irrational
rational
an integer
zero
The Closure Property implies that the product of 45 and 15 is a rational number.
Thanks so much can i ask you some more questions later
You're welcome! Yes, feel free to ask me more questions later. I'm here to help!
How does the Closure Property prove that the sum of a rational and irrational number is irrational?(1 point)
Irrational numbers can be added or subtracted.
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 of an irrational and rational number can’t be predicted.
The Closure Property does not directly prove that the sum of a rational and irrational number is irrational. The Closure Property states that the sum or difference of two rational numbers is a rational number. To prove that the sum of a rational and irrational number is irrational, you would need to provide a separate proof or argument.
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 can’t be simplified, so they must be irrational.
No, radicals always simplify to integers, so they are rational.
Yes, radicals are irrational.
No, some radicals simplify to rational numbers.
No, some radicals simplify to rational numbers.
What type of number will result from the expression 19−−√−3/2?(1 point)
Responses
a rational number
a whole number
an integer
an irrational number
The type of number that will result from the expression 19−−√−3/2 is an irrational number.
What type of number will result from the expression 19−−√−32?(1 point)
Responses
a rational number
a whole number
an integer
an irrational number
The result of the expression 19−√−32 will be an imaginary number.
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 creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer
by choosing random numbers to plug into multiplication problems until he’s convinced
by using a proof by contradiction to show multiplication is a closed operation for rational numbers
He’s correct; it is impossible to prove in any way.
To prove that a rational number multiplied by an irrational number gives an irrational product, one can use a proof by contradiction.
Assume that the product of a rational number (p/q) and an irrational number (r) is rational. Therefore, the result of the multiplication can be expressed as (p/q) * (r) = s, where s is a rational number.
By rearranging the equation, we have r = (s * q) / p. Since s, q, and p are all rational numbers, their product (s * q) / p is also a rational number.
However, we started with the assumption that r is an irrational number. This contradicts our result of (s * q) / p being a rational number.
This contradiction shows that the product of a rational and an irrational number must be an irrational number.
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
1. by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer
2. by choosing random numbers to plug into multiplication problems until he’s convinced
3. by using a proof by contradiction to show multiplication is a closed operation for rational numbers
4. He’s correct; it is impossible to prove in any way.
The correct response is:
1. by creating a proof that shows directly how multiplying a rational and irrational number leads to an irrational answer
Find the product of 12−−√ and 5/6. What type of number is it?(1 point)
Responses
1. 4.1569 . . . ; an irrational number
2. 2.6307 . . . ; an irrational number
3. 2.8867 . . . ; an irrational number
4. 4.2974 . . . ; an irrational number
The product of √12 and 5/6 is approximately 2.8867. Therefore, the correct response is:
3. 2.8867 . . . ; an irrational number
Find the product of 12−−√ and 56. What type of number is it?(1 point)
Responses
4.1569 . . . ; an irrational number
2.6307 . . . ; an irrational number
2.8867 . . . ; an irrational number
4.2974 . . . ; an irrational number
The product of √12 and 56 is approximately 165.529. Therefore, the correct response is not provided in the list. The product is a rational number.
Find the product of 12−−√ and 5/6. What type of number is it?(1 point)
Responses
4.1569 . . . ; an irrational number
2.6307 . . . ; an irrational number
2.8867 . . . ; an irrational number
4.2974 . . . ; an irrational number
The correct response is not provided in the list. The product of √12 and 5/6 is approximately 3.1623. Therefore, the product is a rational number.
What kind of questions help propel discussions?(1 point)
Responses
open-ended questions
questions about people
questions about facts
close-ended questions
The correct response is:
open-ended questions
Which question is open-ended?(1 point)
Responses
Do you like to make art?
What makes a piece of art interesting?
Which artist created this piece of art?
Would you hang this piece of art on your wall?
The open-ended question is:
What makes a piece of art interesting?
What does it mean to show empathy during a disagreement?(1 point)
Responses
It means repeating just the main points of your argument.
It means trying to make your position seem reasonable.
It means trying to understand what others are feeling.
It means explaining an idea to make it clearer.
The correct response is:
It means trying to understand what others are feeling.
Use the scenario to answer the question.
Jess gives a presentation about farm animals. During the presentation, she discusses a lot of different information about each animal. George thinks Jess presented some incorrect information about the nutritional value of cow’s milk. He’s not sure because she also talked about goats at one point.
What question should George ask to clarify what Jess has said?
(1 point)
Responses
Can we look up the information you gave about milk to make sure it’s right?
If you meant cow’s milk earlier, where did you get that information?
Did you know that some of the information you gave about milk is incorrect?
When you talked about the nutritional value of milk, were you talking about cows or goats
The question that George should ask to clarify what Jess has said is:
When you talked about the nutritional value of milk, were you talking about cows or goats?
Use the scenario to answer the question.
Jess gives a presentation about farm animals. During the presentation, she discusses a lot of different information about each animal. George thinks Jess presented some incorrect information about the nutritional value of cow’s milk. He’s not sure because she also talked about goats at one point.
What question should George ask to clarify what Jess has said?
(1 point)
Responses
1. Can we look up the information you gave about milk to make sure it’s right?
2. If you meant cow’s milk earlier, where did you get that information?
3. Did you know that some of the information you gave about milk is incorrect?
4. When you talked about the nutritional value of milk, were you talking about cows or goats
The question that George should ask to clarify what Jess has said is:
4. When you talked about the nutritional value of milk, were you talking about cows or goats?
Use the scenario to answer the question.
Parker explains his thoughts on a new rule the school has implemented. He is outraged because of the impact he thinks it will have—but his friends think he has misunderstood the new rule. They suggest that he ask the school administration if he understands the rule correctly.
What are Parker’s friends asking him to do?
(1 point)
Responses
verify the rule
challenge the rule
discuss the rule
justify the rule
Parker's friends are asking him to:
verify the rule
Rewrite the expression as a product of identical factors to find the product for 100 3/32. Submit your answer in simplified form.
To express 100 3/32 as a product of identical factors, we can rewrite it as:
100 3/32 = (100 * 32 + 3) / 32 = 3200/32 + 3/32
Simplifying further:
3200/32 + 3/32 = 100 + 3/32 = 100 3/32
So, the expression as a product of identical factors is 100 3/32.
Which factors of the base would help simplify the expression 1614?(1 point)
Responses
16=2⋅2⋅2⋅2
16 equals 2 times 2 times 2 times 2
16=8⋅8
16 equals 8 times 8
16=4⋅2⋅2
16 equals 4 times 2 times 2
16=4⋅4
The correct response is:
16 = 4⋅4
Which factors of the base would help simplify the expression 1614?(1 point)
Responses
16=2⋅2⋅2⋅2
16=8⋅8
16=4⋅2⋅2
16=4⋅4
The correct response is:
16 = 4⋅4
Given that the cube root of 27 is 3, what is 27 4/3?(1 point)
Responses
4
12
108
81
The expression 27 4/3 can be simplified as the cube root of 27 raised to the power of 4/3.
Since the cube root of 27 is 3, we have:
(3)^4/3 = 3^(4/3)
The exponent 4/3 refers to taking the cube root and then raising it to the power of 4.
Therefore, 27 4/3 simplifies to 3^(4/3), which equals 27.
So the correct response is:
27
Given that the cube root of 27 is 3, what is 274/3?(1 point)
Responses
4
12
108
81
To simplify the expression 27^(4/3), we can use the fact that when we raise a number to the power of a fraction, it is equivalent to taking the nth root of that number raised to the power of the numerator of the fraction.
In this case, the cube root of 27 is 3. Therefore, we have:
27^(4/3) = (cube root of 27)^4 = 3^4 = 81
So the correct response is:
81
What is the correct simplification of 253/2?(1 point)
Responses
37.5
625
125
75
The correct simplification of 253/2 is:
37.5
What is the correct simplification of 25^3/2?(1 point)
Responses
37.5
625
125
75
The correct simplification of 25^(3/2) is:
125
Given that the meaning of a rational exponent can be connected to the meaning of a root, how can you rewrite 5–√3 using a rational exponent?(1 point)
Responses
51
5 superscript 1 baseline
513
5 superscript Start Fraction 1 over 3 End Fraction baseline
35
3 superscript 5 baseline
53
5 cubed
The correct response is:
5^(1/3)
How can you rewrite 812 using a root?(1 point)
Responses
12−−√8
root index 8 Start Root Start Fraction 1 over 2 End Fraction End Root
8–√1
root index 1 Start Root 8 End Root
8–√12
root index Start Fraction 1 over 2 End Fraction Start Root 8 End Root
8–√
Start Root 8 End Root
The correct response is:
root index 2 Start Root 8 End Root
The correct response is not provided in the list. The correct rewriting of 812 using a root is:
Start Root 8 End Root
How can you rewrite 8^1/2 using a root?(1 point)
Responses
12−−√8
8–√1
8–√12
8–√
The correct response is:
Start Root 8 End Root
How can you rewrite 8^1/2 using a root?(1 point)
Responses
1. 12−−√8
2. 8–√1
3. 8–√12
4. 8–√
How can you rewrite 8^1/2 using a root?(1 point)
Responses
1. 12−−√8
2. 8–√1
3. 8–√12
4. 8–√
What are rhetorical strategies?(1 point)
Responses
the arrangement of words and phrases to create well-formed sentences
statements used to convey meaning, provoke a response, or persuade readers
providing feedback and constructive criticism
the choice and use of words and phrases in a speech or piece of writing