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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

π+4–√
pi plus Start Root 4 End Root

0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction

0.6¯¯¯¯+910

9 answers

To determine which of the sums results in a rational number, let's analyze each option:

  1. \(\sqrt{36} + \sqrt{221}\):

    • \(\sqrt{36} = 6\), which is rational.
    • \(\sqrt{221}\) is not a perfect square, so it is irrational.
    • Therefore, \(6 + \sqrt{221}\) is irrational.

  2. \(\pi + \sqrt{4}\):

    • \(\sqrt{4} = 2\), which is rational.
    • \(\pi\) is an irrational number.
    • Therefore, \(\pi + 2\) is irrational.

  3. \(0.643892553… + \frac{3}{4}\):

    • \(0.643892553...\) appears to be a non-repeating, non-terminating decimal (likely irrational).
    • \(\frac{3}{4}\) is rational.
    • Therefore, \(0.643892553… + \frac{3}{4}\) is irrational.

  4. \(0.6\overline{6} + \frac{9}{10}\):

    • \(0.6\overline{6}\) is a repeating decimal, which is rational (it represents the fraction \(\frac{2}{3}\)).
    • \(\frac{9}{10}\) is rational.
    • The sum of two rational numbers is rational.
    • Therefore, \(0.6\overline{6} + \frac{9}{10}\) is rational.

Based on the analysis, the sum that results in a rational number is:

\(0.6\overline{6} + \frac{9}{10}\).

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, 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+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, 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 integers, and so ad+bcbd is a quotient of two integers.

To determine why \( \frac{ad + bc}{bd} \) is a rational number given that \( a, b, c, \) and \( d \) are integers and \( b \) and \( d \) are non-zero, we need to look at the definitions and properties of rational numbers.

A rational number is defined as a number that can be expressed as the quotient (or fraction) of two integers, where the denominator is not zero.

Let's analyze the responses based on this understanding:

  1. By the Closure Property, \( ad + bc \) and \( bd \) are both quotients of integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.

    • This is incorrect because \( ad + bc \) and \( bd \) are not quotients of integers; rather, they are the results of integer operations.

  2. By the Closure Property, \( ad + bc \) and \( bd \) are both quotients of integers, and so the fraction with numerator \( ad + bc \) and denominator \( bd \) is a quotient of two integers.

    • This is similar to the first option and is also incorrect for the same reasoning.

  3. By the Closure Property, \( ad + bc \) and \( bd \) reduce to a whole number, which is a subset of rational numbers.

    • This statement is misleading since it asserts a reduction to whole numbers without proper justification and doesn't clarify the structure of \( ad + bc \) and \( bd \) as integers.

  4. By the Closure Property, the fraction with numerator \( ad + bc \) and denominator \( bd \) reduces to a whole number, which is a subset of rational numbers.

    • Like the previous option, this one is misleading and does not effectively prove the statement.

  5. By the Closure Property, a quotient of imaginary numbers is a rational number.

    • This is incorrect as it deals with imaginary numbers, which are not relevant in this case since we are dealing with integers.

  6. By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.

    • This is the correct response. Since both \( ad + bc \) and \( bd \) are integers (with \( bd \) being non-zero), the fraction \( \frac{ad + bc}{bd} \) is indeed a valid quotient of two integers, thus forming a rational number.

Given this analysis, the correct explanation is:

By the Closure Property, \( ad + bc \) and \( bd \) are both integers, and so \( \frac{ad + bc}{bd} \) is a quotient of two integers.

The sum of two rational numbers is always(1 point)
Responses

rational.
rational.

irrational.
irrational.

radical.
radical.

zero.

The correct response is:

rational.

The sum of two rational numbers is always rational. This is because rational numbers can be expressed as fractions (where both the numerator and the denominator are integers), and when you add two fractions together, the result is also a fraction, which is a rational number.

The Closure Property implies that the product of 45 and 15 is what type of number?(1 point)
Responses

an integer
an integer

zero
zero

irrational
irrational

rational
rational

The correct response is:

an integer.

The Closure Property states that when you perform certain operations (such as addition or multiplication) on a set of numbers, the result will also be within that same set. Since both 45 and 15 are integers, their product (which is \(45 \times 15 = 675\)) is also an integer.

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

Both options are possible depending on the numbers.
Both options are possible depending on the numbers.

Jolene
Jolene

Malanna
Malanna

Neither of them, it depends on the number.

The correct response is:

Jolene.

When you multiply two rational numbers, the product is always a rational number. Therefore, Jolene is correct in stating that the product of two rational numbers will be rational. Malanna is incorrect because it is not possible for the product of two rational numbers to be irrational.

So, the correct choice is:

Jolene.