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

Question

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

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

π+4–√
pi plus Start Root 4 End Root

0.6¯¯¯¯+9/10

GPT-4o mini · Answer

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

$ \sqrt{36} + \sqrt{221} $
- $ \sqrt{36} = 6 $ (a rational number)
- $ \sqrt{221} $ is not a perfect square, so it is an irrational number.
- The sum $ 6 + \sqrt{221} $ is irrational.
$ 0.643892553... + \frac{3}{4} $
- $ \frac{3}{4} $ is a rational number.
- $ 0.643892553... $ is a decimal which appears to be non-terminating and non-repeating, indicating it is irrational.
- The sum of a rational and an irrational number is irrational.
$ \pi + \sqrt{4} $
- $ \sqrt{4} = 2 $ (a rational number).
- $ \pi $ is an irrational number.
- The sum $ \pi + 2 $ is irrational.
$ 0.6\overline{6} + \frac{9}{10} $
- $ 0.6\overline{6} $ is equal to $ \frac{2}{3} $ (a rational number).
- $ \frac{9}{10} $ is also a rational number.
- The sum $ \frac{2}{3} + \frac{9}{10} $ is rational.

Now, between the two options that result in rational numbers, the ones we found were $ 0.6\overline{6} + \frac{9}{10} $ and $ \frac{3}{4} + 0.643892553...$ (which we concluded is not a rational result).

Thus, the sum that results in a rational number is:

$ 0.6\overline{6} + \frac{9}{10} $

adaleta · Answer

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

GPT-4o mini · Answer

To establish that the expression $\frac{ad + bc}{bd}$ is a rational number, where $a$, $b$, $c$, and $d$ are integers and $b$ and $d$ are non-zero integers, we can reason as follows: 1. **Understanding the components**: - $a$, $b$, $c$, and $d$ are defined as integers, which means they belong to the set of whole numbers, including positive and negative whole numbers and zero (the integers are ...,-3,-2,-1,0,1,2,3,...). - Since both $b$ and $d$ are specifically stated to be non-zero, this guarantees that we will not have division by zero, which is an important condition for our expression to be defined. 2. **Numerator** ($ad + bc$): - The term $ad$ is the product of two integers $a$ and $d$, which results in another integer, as the product of any two integers is also an integer. - Similarly, $bc$ is the product of two integers $b$ and $c$, which also results in an integer. - Therefore, $ad + bc$ is the sum of two integers. The sum of any two integers is also an integer. Thus, $ad + bc$ is an integer. 3. **Denominator** ($bd$): - Since both $b$ and $d$ are non-zero integers, their product $bd$ is also a non-zero integer. The product of two non-zero integers is always a non-zero integer. 4. **Forming the expression**: - We can now consider the entire expression $\frac{ad + bc}{bd}$. - The numerator $ad + bc$ is an integer (let's denote it as $m$), and the denominator $bd$ is a non-zero integer (let's denote it as $n$). 5. **Conclusion**: - By definition, a rational number is any number that can be expressed in the form $\frac{m}{n}$, where $m$ is an integer and $n$ is a non-zero integer. Therefore, since we can express $\frac{ad + bc}{bd}$ in this form with $m = ad + bc$ (an integer) and $n = bd$ (a non-zero integer), we conclude that $\frac{ad + bc}{bd}$ is indeed a rational number. In summary, $\frac{ad + bc}{bd}$ is a rational number because the numerator $ad + bc$ is an integer and the denominator $bd$ is a non-zero integer, thus satisfying the definition of a rational number.

adaleta · Answer

The sum of two rational numbers is always

GPT-4o mini · Answer

The sum of two rational numbers is always a rational number.

To elaborate further, a rational number is defined as any number that can be expressed in the form $\frac{p}{q}$, where $p$ is an integer and $q$ is a non-zero integer.

If you have two rational numbers, say $\frac{a}{b}$ and $\frac{c}{d}$ (where $a$, $b$, $c$, and $d$ are integers and $b$ and $d$ are non-zero), their sum can be computed as follows:

\[ \frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd} \]

Here:

The numerator $ad + bc$ is the sum of two integers (since $a$, $b$, $c$, and $d$ are integers), which is also an integer.
The denominator $bd$ is the product of two non-zero integers, which is also a non-zero integer.

Since the result can be expressed in the form $\frac{p}{q}$ with $p = ad + bc$ (an integer) and $q = bd$ (a non-zero integer), the sum is indeed a rational number.

Therefore, the conclusion is that the sum of two rational numbers is always a rational number.

adaleta · Answer

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

GPT-4o mini · Answer

The Closure Property states that if you perform a specific operation (such as addition or multiplication) on elements of a set and the result is also an element of that set, then the set is said to be closed under that operation.

In this case, you are multiplying $45$ and $15$:

\[ 45 \times 15 = 675 \]

Both $45$ and $15$ are integers. Since the product of two integers (in this case, $675$) is also an integer, we can conclude that the set of integers is closed under multiplication.

Therefore, the product of $45$ and $15$ is an integer.

adaleta · Answer

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

GPT-4o mini · Answer

To analyze the product of $\frac{4}{5}$ and $15$: $$ \frac{4}{5} \times 15 $$ Calculating this: $$ \frac{4 \times 15}{5} = \frac{60}{5} = 12 $$ Now, the Closure Property indicates that if you perform an operation (like multiplication) on elements of a set (in this case, rational numbers) and the result is also an element of that set, then the set is considered closed under that operation. 1. **Types of Numbers**: - $\frac{4}{5}$ is a **rational number**. - $15$ is also a **rational number** (it can be expressed as $\frac{15}{1}$). - The product $12$ is an integer (which is also a rational number since all integers can be expressed as a fraction with a denominator of $1$). Since both $\frac{4}{5}$ and $15$ are rational numbers and their product $12$ is also a rational number, we can say that the set of rational numbers is closed under multiplication. Thus, the product of $\frac{4}{5}$ and $15$ is a **rational number**.

adaleta · Answer

Question 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)

GPT-4o mini · Answer

Jolene is correct.

When you multiply two rational numbers, the product will always be a rational number.

Explanation:

A rational number is any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.
If you have two rational numbers, say $\frac{a}{b}$ and $\frac{c}{d}$ (where $a$, $b$, $c$, and $d$ are integers and $b$ and $d$ are non-zero), their product is calculated as follows:

\[ \frac{a}{b} \times \frac{c}{d} = \frac{a \cdot c}{b \cdot d} \]

Since $a \cdot c$ is an integer and $b \cdot d$ is a non-zero integer, the result $\frac{a \cdot c}{b \cdot d}$ is also a rational number.

Therefore, when multiplying two rational numbers, the product is always a rational number, which means Malanna's assertion that the product will be irrational is incorrect.