According to the Closure Property, which of these sums results in a rational number?(1 point)

Question

Responses

36−−√+221−−−√
Start Root 36 End Root plus Start Root 221 End Root

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

0.6¯¯¯¯+9/10
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction

π+4–√

In the proof that a/b+c/d
, 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/bd
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 quotients of integers, and so ad+bc/bd
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+bc/bd
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, a quotient of imaginary numbers is a rational number.

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

rational.
rational.

zero.
zero.

radical.
radical.

irrational.

The Closure Property implies that the product of 4/5
and 15 is what type of number?(1 point)
Responses

irrational
irrational

rational
rational

an integer
an integer

zero

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

Jolene
Jolene

Neither of them, it depends on the number.
Neither of them, it depends on the number.

Malanna
Malanna

Both options are possible depending on the numbers.

Answer 1

Here are the responses to the questions:

1. According to the Closure Property, which of these sums results in a rational number?

   The correct response is 0.6¯¯¯¯ + 9/10, because:

   • \( 0.6\overline{6} \) (which equals \( \frac{2}{3} \)) is a rational number, and \( \frac{9}{10} \) is also rational. The sum of two rational numbers is rational.

2. In the proof that \( \frac{a}{b} + \frac{c}{d} \), where a, b, c, and d are integers and b and d are non-zero, explain why \( \frac{ad + bc}{bd} \) is a rational number.

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

3. The sum of two rational numbers is always:

   The correct response is rational.

4. The Closure Property implies that the product of \( \frac{4}{5} \) and 15 is what type of number?

   The correct response is rational, because the product of a rational number and an integer is always rational.

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?

   The correct response is Jolene, because the product of two rational numbers is always rational.

Let me know if you need further clarification on any of the questions!