In the proof that the sum of rational numbers ab

Question

𝑎
𝑏
and cd
𝑐
𝑑
, where a, b, c, and d are integers and b and d are non-zero, is a rational number, name the property that states that ad+bc
𝑎
𝑑
+
𝑏
𝑐
and bd
𝑏
𝑑
are integers. Enter 1 for option 1, 2 for option 2, and 3 for option 3.

Option 1: Associative Property

Option 2: Closure Property

Option 3: Commutative Property



(1 point)

Answer 1

The property that states that \( ad + bc \) and \( bd \) are integers is the Closure Property. The Closure Property indicates that when you perform an operation (in this case, addition and multiplication) on integers, the result is also an integer.

So, the correct answer is:

Option 2: Closure Property

Enter 2.

Answer 2

Without solving, will the sum of 4.2+36−−√
4.2
+
36
be rational or irrational? Enter 1 for rational. Enter 2 for irrational. (1 point)

Answer 3

To determine whether the sum \( 4.2 + \sqrt{36} \) is rational or irrational, we need to analyze the components:

• \( 4.2 \) is a rational number (it can be expressed as a fraction, \( \frac{42}{10} \)).
• \( \sqrt{36} = 6 \), which is also a rational number.

The sum of two rational numbers is always rational. Therefore, \( 4.2 + 6 \) will be rational.

So, the correct answer is:

Enter 1 for rational.

Answer 4

What must be true about ab
𝑎
𝑏
if cd
𝑐
𝑑
and ad+bcbd
𝑎
𝑑
+
𝑏
𝑐
𝑏
𝑑
are both rational numbers? Enter 1 for option 1 and 2 for option 2.

Option 1: ab
𝑎
𝑏
is rational

Option 2: ab
𝑎
𝑏
is irrational

(1 point)

Answer 5

If \( \frac{c}{d} \) and \( \frac{ad + bc}{bd} \) are both rational numbers, this implies that \( ab \) must also be rational. This is because the operation of addition and division of rational numbers always results in a rational number, and since \( \frac{a}{b} \) is involved in the definition of \( ab \), it must be rational as well.

Thus, the correct choice is:

Option 1: \( \frac{a}{b} \) is rational.

Enter 1.

Answer 6

Without solving, will the product of 49−−√
49
and 13
1
3
be rational or irrational? Enter 1 for rational. Enter 2 for irrational.(1 point)

Answer 7

To determine whether the product \( \sqrt{49} \cdot 13 \) is rational or irrational, we evaluate the components:

• \( \sqrt{49} = 7 \), which is a rational number.
• \( 13 \) is also a rational number.

The product of two rational numbers is always rational. Therefore, \( 7 \cdot 13 \) will be rational.

So, the correct answer is:

Enter 1 for rational.

Answer 8

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

0.6¯¯¯¯+910
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction

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

Answer 9

To determine which of the given sums results in a rational number according to the Closure Property, let's analyze each option:

1. \( 0.6\overline{6} + \frac{9}{10} \):

   • \( 0.6\overline{6} \) is equivalent to the rational number \( \frac{2}{3} \).
   • The sum \( \frac{2}{3} + \frac{9}{10} \) is a rational number.

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

   • \( \sqrt{36} = 6 \) is a rational number, but \( \sqrt{221} \) is irrational.
   • The sum of a rational number and an irrational number is irrational.

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

   • \( \sqrt{4} = 2 \) is a rational number, but \( \pi \) is irrational.
   • The sum of a rational number and an irrational number is irrational.

4. \( 0.643892553\ldots + 34 \):

   • The number \( 0.643892553\ldots \) seems to be an approximate decimal representation, likely representing an irrational number.
   • However, since 34 is a whole number (rational), if \( 0.643892553\ldots \) is indeed an approximation or an irrational number, the sum will still be irrational.

Based on the above evaluations, the only sum that results in a rational number is:

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

So, the answer is:

Choose the first option.

Answer 10

In the proof that ab+cd
𝑎
𝑏
+
𝑐
𝑑
, where a, b, c, and d are integers and b and d are non-zero, explain why ad+bcbd
𝑎
𝑑
+
𝑏
𝑐
𝑏
𝑑
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
𝑎
𝑑
+
𝑏
𝑐
and bd
𝑏
𝑑
are both integers, and so ad+bcbd
𝑎
𝑑
+
𝑏
𝑐
𝑏
𝑑
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+bc
𝑎
𝑑
+
𝑏
𝑐
and bd
𝑏
𝑑
are both quotients of integers, and so ad+bcbd
𝑎
𝑑
+
𝑏
𝑐
𝑏
𝑑
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
𝑎
𝑑
+
𝑏
𝑐
𝑏
𝑑
reduces to a whole number, which is a subset of rational numbers.

Answer 11

To explain why \( \frac{ad + bc}{bd} \) is a rational number in the proof that \( \frac{a}{b} + \frac{c}{d} \) is rational, we need to recall the definition of rational numbers. A number is rational if it can be expressed as a quotient of two integers, where the denominator is not zero.

Here's the relevant reasoning based on the closure property:

• \( ad + bc \) is an integer because both \( a, b, c, d \) are integers, and the sum and product of integers are integers.
• \( bd \) is also an integer, where both \( b \) and \( d \) are non-zero integers; hence, the product is also non-zero.

Thus, the expression \( \frac{ad + bc}{bd} \) consists of an integer \( (ad + bc) \) as the numerator and a non-zero integer \( (bd) \) as the denominator, making it a quotient of two integers.

The best response would be:

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

This option correctly encapsulates the reasoning behind why \( \frac{ad + bc}{bd} \) is a rational number.