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# Common Core Standard HSN-RN.B.3 Questions

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

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Grade 11 Rational and Irrational Numbers CCSS: HSN-RN.B.3

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What is the missing reason in step 2?
1. Given
2. Division Property of Equality
3. Definition of rational numbers
4. Any number can be rewritten in another form
Grade 10 Rational and Irrational Numbers CCSS: HSN-RN.B.3
Which of the following, if added to 10, would result in a rational number? Choose all correct answers.
1. $sqrt(3)$
2. $sqrt(16)$
3. $pi$
4. $4/3$
Grade 10 Rational and Irrational Numbers CCSS: HSN-RN.B.3
Which of the following, if multiplied by the number 25, would result in a rational number? Choose all that apply.
1. $sqrt(5)$
2. $e^2$
3. $13.000004$
4. $-1$
Grade 11 Rational and Irrational Numbers CCSS: HSN-RN.B.3
Andy wants to prove that the sum of a rational number and irrational number is irrational. He does this by the following.

Given $x$ is a rational number and $y$ is an irrational number. Assume that $x+y$ is rational, and let the sum be represented by $w$. Since any rational number can be represented as the division of two integers, let $x=a/b$ and $w=c/d$, where $a,b,c,d$ are all integers. Therefore, $x+y=w$ can be written as $a/b + y = c/d$. By the subtraction rule of equality, $y = c/d - a/b$. Rewriting the difference of the two fractions (using the multiplicative identity and distributive property), this becomes $y = (cb - ad)/(db)$. Since integers are closed under subtraction and multiplication, $cb-ad$ is an integer, and so is $db$. Therefore, this would make $y$ a rational number, by definition of rational numbers. But this contradicts our given, that $y$ is irrational. Therefore, $x+y$ must be irrational.

Is this proof correct? If not, why?
1. Yes, this proof is correct. It is called a proof by contradiction.
2. No, it is not correct. Andy has to use a two-table format for this to be a valid proof.
3. No, it is not correct. This only proves that $x+y$ is not rational, but that doesn't necessarily mean it is irrational.
4. No, it is not correct. Andy cannot prove something by contradiction; he must directly prove it.
Grade 11 Rational and Irrational Numbers CCSS: HSN-RN.B.3
Which of the following disproves the statement that the sum of two irrational numbers is always irrational?
1. $sqrt(3) + sqrt(3) = 6$
2. $sqrt(2) + (-sqrt(2)) = 0$
3. $sqrt(16) + sqrt(4) = 6$
4. None of the above, since this is a true statement.
Grade 11 Rational and Irrational Numbers CCSS: HSN-RN.B.3
Which of the following disprove(s) the statement that the product of two irrational numbers is always irrational? Choose all that apply.
1. $sqrt(5) * (1/sqrt(5)) = 1$
2. $sqrt(9) * 1/root{3}{27} = 1$
3. $1/sqrt(2) * sqrt(8) = 2$
4. None of the above, since this statement is true.
Grade 10 Rational and Irrational Numbers CCSS: HSN-RN.B.3
Let $x=sqrt(3)$. Which of the following results in an irrational number? Choose all that apply.
1. $3x$
2. $x/0$
3. $x+9$
4. $x+sqrt(x)$
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