In the proof that the sum of rational numbers ab and cd , 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 and bd are integers. Enter 1 for option 1, 2 for option 2, and 3 for option 3.

Question

In the proof that the sum of rational numbers  ab  and  cd , 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  and  bd  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

GPT-4o mini · Answer

The property that states that $ ad + bc $ and $ bd $ are integers is the **Closure Property**. Therefore, the correct answer is: **2**

cocopuff · Answer

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

GPT-4o mini · Answer

To determine if the sum $ 4.2 + \sqrt{36} $ is rational or irrational, we first note that $ \sqrt{36} = 6 $, which is a rational number. The number $ 4.2 $ is also rational. The sum of two rational numbers is always rational. Therefore, $ 4.2 + \sqrt{36} = 4.2 + 6 $ will be rational. The correct answer is: **1**

cocopuff · Answer

What must be true about ab if cd and ad+bcbd are both rational numbers? Enter 1 for option 1 and 2 for option 2. Option 1: ab is rational Option 2: ab is irrational (1 point)

GPT-4o mini · Answer

If $ cd $ and $ \frac{ad + bc}{bd} $ are both rational numbers, then for $ ab $ to maintain the overall property of rationals, it must also be rational. Thus, if $ cd $ is rational (which implies $ c $ and $ d $ are such that their product is a rational number) and $ \frac{ad + bc}{bd} $ is also rational (which involves the integers $ a, b, c, d $), it directly follows that $ ab $ must also be rational. Therefore, the correct answer is: **1**

cocopuff · Answer

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

GPT-4o mini · Answer

The product of $ \sqrt{49} $ and $ 13 $ can be determined as follows: Since $ \sqrt{49} = 7 $, which is a rational number, and $ 13 $ is also a rational number, the product of two rational numbers is always rational. Therefore, the product $ \sqrt{49} \times 13 = 7 \times 13 $ will be rational. The correct answer is: **1**

cocopuff · Answer

In the proof that the product of ab and cd , where a, b, c, and d are integers and b and d are non-zero, what does the Closure Property tell you about the products a⋅c and b⋅d ? Enter 1 for option 1 and 2 for option 2.

Option 1: They are both imaginary numbers.

Option 2: They are both integers.

(1 point)

GPT-4o mini · Answer

The Closure Property states that the product of two integers is always an integer. Since $ a $, $ b $, $ c $, and $ d $ are all integers, the products $ a \cdot c $ and $ b \cdot d $ must also be integers.

Therefore, the correct answer is:

2