Prove that the product of the two rational numbers 36−−√

Question

Prove that the product of the two rational numbers  36−−√
  and  5/3
  is a rational number. Find the product and explain how you know it is a rational number. Explain your reasoning in 1–2 sentences.

GPT-4o mini · Answer

To prove that the product of $ \sqrt{36} $ and $ \frac{5}{3} $ is a rational number, we first calculate the product. $$ \sqrt{36} = 6 $$ Now, we find the product: $$ 6 \times \frac{5}{3} = \frac{6 \times 5}{3} = \frac{30}{3} = 10 $$ Since $ 10 $ is a whole number, it is also a rational number (all integers are rational as they can be expressed as a fraction with a denominator of 1). Thus, the product of $ \sqrt{36} $ and $ \frac{5}{3} $ is a rational number.

adaleta · Answer

please in simple sentences

GPT-4o mini · Answer

The square root of 36 is 6. When we multiply 6 by $ \frac{5}{3} $, we get $ 10 $, which is a whole number. Since 10 can be written as $ \frac{10}{1} $, it is a rational number. Therefore, the product of $ \sqrt{36} $ and $ \frac{5}{3} $ is a rational number.

adaleta · 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.

GPT-4o mini · Answer

To prove that the sum of $ 34 $ and $ \sqrt{10} $ is irrational, we can assume the opposite — that their sum is rational — and reach a contradiction. 1. **Assume the Sum is Rational**: Let’s assume $ 34 + \sqrt{10} $ is rational. This means $ 34 + \sqrt{10} = r $ for some rational number $ r $. 2. **Isolate the Square Root**: Rearranging gives us: $$ \sqrt{10} = r - 34 $$ Since $ r $ is rational and $ 34 $ is also rational, the difference $ r - 34 $ is rational. 3. **Rationality of the Square Root**: This implies that $ \sqrt{10} $ is rational (as it equals a rational number). 4. **Contradiction**: However, it is known that $ \sqrt{10} $ is an irrational number (because 10 is not a perfect square), which contradicts our assumption that $ 34 + \sqrt{10} $ is rational. Therefore, since our assumption led to a contradiction, we conclude that $ 34 + \sqrt{10} $ must be irrational. ### Explanation: The sum $ 34 + \sqrt{10} $ is irrational because it combines a rational number (34) with an irrational number ($ \sqrt{10} $), and the sum of a rational and an irrational number is always irrational.

adaleta · Answer

Prove that the sum of 3/4 and 10−−√ is irrational. Show your work, and in 1–2 sentences, explain why the sum is an irrational number.

GPT-4o mini · Answer

To prove that the sum of $ \frac{3}{4} $ and $ \sqrt{10} $ is irrational, we will assume the opposite and arrive at a contradiction. 1. **Assume the Sum is Rational**: Let’s assume $ \frac{3}{4} + \sqrt{10} $ is rational. This means there exists a rational number $ r $ such that: $$ \frac{3}{4} + \sqrt{10} = r $$ 2. **Isolate the Square Root**: Rearranging gives us: $$ \sqrt{10} = r - \frac{3}{4} $$ Since $ r $ is rational and $ \frac{3}{4} $ is also rational, the difference $ r - \frac{3}{4} $ is rational. 3. **Rationality of the Square Root**: This means that $ \sqrt{10} $ is rational, as it equals a rational number. 4. **Contradiction**: However, it is known that $ \sqrt{10} $ is irrational (because 10 is not a perfect square), which contradicts our previous conclusion that $ \sqrt{10} $ can be expressed as a rational number. Therefore, since our assumption leads to a contradiction, we conclude that $ \frac{3}{4} + \sqrt{10} $ must be irrational. ### Explanation: The sum $ \frac{3}{4} + \sqrt{10} $ is irrational because it adds a rational number ($\frac{3}{4}$) to an irrational number ($\sqrt{10}$), and the sum of a rational and an irrational number is always irrational.

adaleta · Answer

what is the sum

GPT-4o mini · Answer

To find the sum of $ \frac{3}{4} $ and $ \sqrt{10} $, we can express it as follows: $$ \frac{3}{4} + \sqrt{10} $$ Since $ \sqrt{10} $ is approximately $ 3.162 $, we can compute the sum numerically: 1. Convert $ \frac{3}{4} $ to a decimal: $$ \frac{3}{4} = 0.75 $$ 2. Add it to the approximate value of $ \sqrt{10} $: $$ 0.75 + 3.162 \approx 3.912 $$ Thus, the sum is: $$ \frac{3}{4} + \sqrt{10} \approx 4.912 $$ However, the exact sum remains $ \frac{3}{4} + \sqrt{10} $, which cannot be simplified further without a decimal approximation.

adaleta · Answer

Prove that the product of 2π and 3/4 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 $ \frac{3}{4} $ is an irrational number, we can start by calculating the product: $$ 2\pi \times \frac{3}{4} = \frac{3 \times 2\pi}{4} = \frac{6\pi}{4} = \frac{3\pi}{2} $$ ### Explanation of Irrationality: We know that $ \pi $ is an irrational number. Since the product of a non-zero rational number ($ \frac{3}{2} $) and an irrational number ($ \pi $) is always irrational, the product $ \frac{3\pi}{2} $ is also irrational. ### Reasoning: Thus, $ 2\pi \times \frac{3}{4} = \frac{3\pi}{2} $ is irrational because multiplying an irrational number by a non-zero rational number results in an irrational number.