Evaluate the two sums and determine which statement is true. Enter the correct number associated with your response.

Question

Column A	Column B
0.5+34

3–√+25

Statement #1: Only Column A has a rational sum.

Statement #2: Only Column B has a rational sum.

Statement #3: Column A and Column B both have rational sums.

(1 point)
The simplified values of both columns indicate that Statement #
 is true.

GPT-4o mini · Answer

Let's evaluate the two columns: **Column A:** $$ 0.5 + 34 = 34.5 $$ 34.5 is a rational number since it can be expressed as $ \frac{69}{2} $. **Column B:** $$ 3 - \sqrt{25} = 3 - 5 = -2 $$ -2 is also a rational number since it can be expressed as $ \frac{-2}{1} $. Now let's compare the two columns: - Column A (34.5) is rational. - Column B (-2) is rational. Both Column A and Column B have rational sums. Therefore, the correct statement is: **Statement #3: Column A and Column B both have rational sums.** So, the answer is that Statement #3 is true.

cocopuff · Answer

The product of a multiplication problem is 225−−−√ . What do you know about the factors?(1 point)
Responses

One factor is rational, but the other is irrational.
One factor is rational, but the other is irrational.

At least one factor is irrational because the product is irrational.
At least one factor is irrational because the product is irrational.

It cannot be predicted based on the information given.
It cannot be predicted based on the information given.

They are both rational because the product is rational.

GPT-4o mini · Answer

To analyze the situation, let's first consider the product you're dealing with, which is $ \sqrt{225} $.

Calculating the square root:

\[ \sqrt{225} = 15 \]

15 is a rational number.

Now, let's think about the factors of 15. The factors of a number can be either rational or irrational. However, since the product (15 in this case) is rational, we can conclude something about the possible factors:

If both factors are rational, their product is certainly rational.
If one factor is rational and the other is irrational, their product would be irrational.
If both factors are irrational, their product could be rational or irrational, depending on the specific numbers.

Since we have established that the product is rational (15), it follows that:

Both factors must be rational, or
Both factors could be rational while having irrational components, as long as their overall product yields a rational number.

Given the context of standard multiplication, since the product is rational and assuming we are not dealing with complexity like fractional components that yield an overall rational count, it leads us to conclude:

They are both rational because the product is rational.

So the correct response is: They are both rational because the product is rational.

cocopuff · Answer

Put the steps to the proof that the sum t=r+s is irrational if r is a rational and s is rational in the correct sequence.

Step 1. Subtract, writing s as a fraction.

Step 2. For a contradiction, assume that t is rational, and write r and t as fractions.

Step 3. Realize a contradiction.

Step 4. Find a common denominator.

(1 point)
Responses

Step 3, Step 4, Step 2, Step 1
Step 3, Step 4, Step 2, Step 1

Step 2, Step 4, Step 1, Step 3
Step 2, Step 4, Step 1, Step 3

Step 1, Step 3, Step 2, Step 4
Step 1, Step 3, Step 2, Step 4

Step 4, Step 2, Step 1, Step 3

GPT-4o mini · Answer

To prove that the sum $ t = r + s $ is irrational if both $ r $ and $ s $ are rational, we can follow these logical steps: 1. **Step 2**: For a contradiction, assume that $ t $ is rational, and write $ r $ and $ t $ as fractions. This gives the foundation of the proof by contradiction. 2. **Step 4**: Find a common denominator. You need this step to combine the fractions in the next step. 3. **Step 1**: Subtract, writing $ s $ as a fraction. Rewrite $ t = r + s $ as $ s = t - r $, and substitute the fractions from the previous steps. 4. **Step 3**: Realize a contradiction. This final step will show that the assumption that $ t $ is rational leads to a contradiction based on the rational/irrational nature of $ s $. Thus, the correct sequence of steps is: **Step 2, Step 4, Step 1, Step 3**. So the answer is: **Step 2, Step 4, Step 1, Step 3**.

cocopuff · Answer

Charles, Zayeer, and Kali are trying to simplify 1,00023 . Charles says the correct simplification is 10 because 1,000=10⋅10⋅10 and 1,00023=10 . Zayeer says the correct simplification is 1,000 because 1,000=10⋅10⋅10→1,00023=10⋅10⋅10=1,000 . Kali says the correct simplification is 100 because 1,000=10⋅10⋅10→1,00023=10⋅10=100 . Who has the correct value?(1 point)
Responses

Charles
Charles

Kali
Kali

Zayeer
Zayeer

none of them

GPT-4o mini · Answer

To simplify $ 1,000^{23} $, we start with the basic fact that $ 1,000 = 10^3 $. Therefore, we can rewrite $ 1,000^{23} $ using exponent rules: $$ 1,000^{23} = (10^3)^{23} $$ Using the power of a power property of exponents (which states $ (a^m)^n = a^{m \cdot n} $), we have: $$ (10^3)^{23} = 10^{3 \cdot 23} = 10^{69} $$ Now, let's evaluate what each person stated: 1. **Charles** says the correct simplification is 10. He is incorrect because $ 1,000^{23} = 10^{69} $, not just 10. 2. **Zayeer** says the correct simplification is 1,000. He is also incorrect because $ 1,000^{23} = 10^{69}$, not just 1,000. 3. **Kali** says the correct simplification is 100. She is likewise incorrect, for the same reasons as above. None of these interpretations reach the correct simplification of $ 1,000^{23} = 10^{69} $. Thus, the answer is: **none of them**.

cocopuff · Answer

What is the simplest form of 2512 ?(1 point) Responses 5 5 25−−√ Start Root 25 End Root 12.5 12.5 25−−√12

GPT-4o mini · Answer

To find the simplest form of $ 25^{\frac{1}{2}} $, we will evaluate $ 25^{\frac{1}{2}} $. Since $ 25 = 5^2 $: $$ 25^{\frac{1}{2}} = \sqrt{25} = \sqrt{5^2} = 5 $$ Therefore, the simplest form of $ 25^{\frac{1}{2}} $ is $ 5 $. So, the correct answer is: **5**.

cocopuff · Answer

Consider the expression 473+121−−−√ . Will the sum of the two rational numbers produce a rational number? Explain your reasoning in 1–2 sentences.(2 points)

GPT-4o mini · Answer

Yes, the expression $ 473 + \sqrt{121} $ will produce a rational number. This is because $ \sqrt{121} = 11 $, which is a rational number, and the sum of two rational numbers ($ 473 + 11 $) is always rational. Thus, $ 473 + 11 = 484 $, which is rational.

cocopuff · Answer

Prove that the product of the two rational numbers 36−−√ and 53 is a rational number. Find the product and explain how you know it is a rational number. Explain your reasoning in 1–2 sentences.(2 point

GPT-4o mini · Answer

To prove that the product of the two rational numbers $ \sqrt{36} $ and $ 53 $ is a rational number, we first simplify $ \sqrt{36} $:

\[ \sqrt{36} = 6 \]

Now, we can calculate the product:

\[ 6 \times 53 = 318 \]

Since both $ 6 $ and $ 53 $ are rational numbers (as they can be expressed as $ \frac{6}{1} $ and $ \frac{53}{1} $), and the product of two rational numbers is always rational, we conclude that $ 318 $ is a rational number. Therefore, the product $ 6 \times 53 = 318 $ confirms that the product is rational.

cocopuff · Answer

Prove that the sum of 34 and 10−−√ is irrational. Show your work, and in 1–2 sentences, explain why the sum is an irrational number.(2 points)

GPT-4o mini · Answer

To prove that the sum $ 34 + \sqrt{10} $ is irrational, we start by recognizing that $ 34 $ is a rational number, as it can be expressed as $ \frac{34}{1} $. However, $ \sqrt{10} $ is an irrational number because it cannot be expressed as a fraction of two integers. Now we will assume for the sake of contradiction that the sum $ 34 + \sqrt{10} $ is rational. If it were rational, let's denote it as $ r $: $$ r = 34 + \sqrt{10} $$ Rearranging this gives us: $$ \sqrt{10} = r - 34 $$ Since $ r $ is rational and $ 34 $ is also rational, their difference $ (r - 34) $ would also be rational. This implies that $ \sqrt{10} $ would be rational, which contradicts the established fact that $ \sqrt{10} $ is indeed irrational. Therefore, our assumption that $ 34 + \sqrt{10} $ is rational must be false, leading us to conclude that $ 34 + \sqrt{10} $ is irrational. In summary, the sum $ 34 + \sqrt{10} $ is irrational because it is the sum of a rational number (34) and an irrational number ($\sqrt{10}$), and the sum of a rational and an irrational number is always irrational.

cocopuff · Answer

Prove that the product of 2π and 34 is an irrational number. Find the product and explain why the value is irrational. Explain your reasoning in 1–2 sentences.(2 points)

GPT-4o mini · Answer

To prove that the product of $ 2\pi $ and $ 34 $ is an irrational number, we first calculate the product: $$ 2\pi \times 34 = 68\pi $$ Next, we note that $ \pi $ is an irrational number. A fundamental property of irrational numbers is that the product of a non-zero rational number and an irrational number is always irrational. Since $ 34 $ is a rational number and $ 2\pi $ includes $ \pi $, it follows that $ 68\pi $ must also be irrational. Thus, the reasoning is that $ 68\pi $ is irrational because it is the product of a rational number (34) and an irrational number ($ 2\pi $), which results in an irrational number. The calculated product $ 68\pi $ confirms this conclusion, as it cannot be expressed as a fraction of two integers.