Prove that the product of the two rational numbers sqrt(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.

Question

GPT-4o mini · Answer

To prove that the product of $\sqrt{36}$ and $\frac{5}{3}$ is a rational number, we first calculate $\sqrt{36} = 6$, which is a rational number since it can be expressed as $\frac{6}{1}$. Now, we find the product: $$ 6 \times \frac{5}{3} = \frac{6 \times 5}{3} = \frac{30}{3} = 10. $$ Since $10$ can also be expressed as a fraction $\frac{10}{1}$, it is a rational number. Therefore, the product of $\sqrt{36}$ and $\frac{5}{3}$ is rational because it is expressed as the ratio of two integers (30 and 3).

Answer

Prove that the sum of 3/4 and sqrt(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 $ \frac{3}{4} + \sqrt{10} $ is irrational, we will use a proof by contradiction. Assume that the sum $ x = \frac{3}{4} + \sqrt{10} $ is rational. By definition, a rational number can be expressed as the ratio of two integers $ p/q $ where $ p $ and $ q $ are integers and $ q \neq 0 $. From our assumption: $$ \sqrt{10} = x - \frac{3}{4}. $$ Since $ x $ is rational and $ \frac{3}{4} $ is rational, the right-hand side $ x - \frac{3}{4} $ is also rational. This means that $ \sqrt{10} $ is rational as well. However, $ \sqrt{10} $ is known to be an irrational number since it cannot be expressed as a fraction of two integers. Thus, we reach a contradiction where we assumed that a rational number led to the conclusion that $ \sqrt{10} $ is rational. Therefore, our original assumption that $ \frac{3}{4} + \sqrt{10} $ is rational must be false. Thus, the sum $ \frac{3}{4} + \sqrt{10} $ is irrational. In summary, the sum is irrational because assuming it to be rational leads to the false conclusion that $ \sqrt{10} $ is rational, which contradicts the established knowledge that $ \sqrt{10} $ is irrational.

Answer

Can you make it shorter

GPT-4o mini · Answer

To prove that $ \frac{3}{4} + \sqrt{10} $ is irrational, assume it is rational. Then, we can express it as $ x = \frac{3}{4} + \sqrt{10} $, leading to $ \sqrt{10} = x - \frac{3}{4} $. Since both $ x $ and $ \frac{3}{4} $ are rational, their difference $ \sqrt{10} $ would also have to be rational. However, $ \sqrt{10} $ is known to be irrational, leading to a contradiction. Thus, $ \frac{3}{4} + \sqrt{10} $ is irrational because it results in the false conclusion that an irrational number is rational.

Answer

can you make it look easier

GPT-4o mini · Answer

Sure! Here’s a simplified version:

Assume $ \frac{3}{4} + \sqrt{10} $ is rational. This means we can write it as a fraction. If we rearrange, we find:

\[ \sqrt{10} = \left( \frac{3}{4} + \sqrt{10} \right) - \frac{3}{4}. \]

Since both $ \frac{3}{4} $ and $ \left( \frac{3}{4} + \sqrt{10} \right) $ are rational, their difference $ \sqrt{10} $ must also be rational. However, $ \sqrt{10} $ is known to be irrational, so our assumption is wrong.

Therefore, $ \frac{3}{4} + \sqrt{10} $ is irrational because it leads to a contradiction that an irrational number is rational.

Answer

what does the left and right indicate

GPT-4o mini · Answer

In the equation $$ \sqrt{10} = \left( \frac{3}{4} + \sqrt{10} \right) - \frac{3}{4}, $$ the left side, $ \sqrt{10} $, represents the expression we are analyzing. The right side shows the assumption that if $ \frac{3}{4} + \sqrt{10} $ is rational (can be expressed as a fraction), then subtracting the rational number $ \frac{3}{4} $ from it should also yield a rational number. This leads us to conclude that $ \sqrt{10} $ would then also have to be rational. The contradiction arises because we know $ \sqrt{10} $ is irrational, which shows that our initial assumption—that $ \frac{3}{4} + \sqrt{10} $ is rational—must be incorrect. Thus, it indicates that $ \frac{3}{4} + \sqrt{10} $ is irrational.