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.(2 points)

Answer 1

To prove that the product of \(\sqrt{36}\) and \(\frac{5}{3}\) is a rational number, we first calculate \(\sqrt{36}\), which equals \(6\). Now, we find the product:

\[ 6 \times \frac{5}{3} = \frac{6 \times 5}{3} = \frac{30}{3} = 10 \]

Since both \(6\) and \(\frac{5}{3}\) are rational numbers, their product is also a rational number, as the set of rational numbers is closed under multiplication. Thus, we conclude that the product \(10\) is a rational number.

Answer 2

Can you please explain this in writing?

Answer 3

Certainly!

To demonstrate that the product of the two rational numbers \(\sqrt{36}\) and \(\frac{5}{3}\) is a rational number, we start by evaluating \(\sqrt{36}\). The square root of \(36\) is \(6\), which is a rational number.

Next, we compute the product of \(6\) and \(\frac{5}{3}\):

\[ 6 \times \frac{5}{3} = \frac{6 \times 5}{3} = \frac{30}{3} = 10 \]

The result of this multiplication is \(10\), which is also a rational number.

We know that the product of two rational numbers is always a rational number. This is because a rational number can be expressed as the quotient of two integers, and multiplying two such quotients results in another quotient of two integers.

In conclusion, since both \(\sqrt{36}\) (which is \(6\)) and \(\frac{5}{3}\) are rational numbers, their product \(10\) is also a rational number.

Answer 4

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.

Answer 5

To prove that the sum of \(\frac{3}{4}\) and \(\sqrt{10}\) is irrational, we can use the property that the sum of a rational number and an irrational number is always irrational.

1. Identify the numbers involved:

   • \(\frac{3}{4}\) is a rational number because it can be expressed as the fraction of two integers (3 and 4).
   • \(\sqrt{10}\) is an irrational number. An irrational number cannot be expressed as a fraction of two integers. To prove that \(\sqrt{10}\) is irrational, one can assume the opposite (that it is rational) and show a contradiction. For instance, if \(\sqrt{10} = \frac{a}{b}\) (where \(a\) and \(b\) are coprime integers), squaring both sides leads to \(10 = \frac{a^2}{b^2}\) or \(a^2 = 10b^2\). This means \(a^2\) is divisible by 10, implying \(a\) is also divisible by 10. Letting \(a = 10k\) leads to \(10k^2 = b^2\), indicating \(b^2\) is also divisible by 10, which contradicts the assumption that \(a\) and \(b\) are coprime — therefore, \(\sqrt{10}\) is irrational.

2. Compute the sum: \[ \frac{3}{4} + \sqrt{10} \]

3. Show that the sum is irrational: Assume for contradiction that \(\frac{3}{4} + \sqrt{10}\) is rational. Then, we can rearrange this to find \(\sqrt{10}\):

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

   Since \(\frac{3}{4}\) is rational and we are assuming \(\frac{3}{4} + \sqrt{10}\) is rational, this implies \(\sqrt{10}\) would also be rational (because the difference between two rational numbers is rational). However, this contradicts the earlier established fact that \(\sqrt{10}\) is irrational.

Conclusion: The assumption that \(\frac{3}{4} + \sqrt{10}\) is rational leads to a contradiction. Therefore, the sum \(\frac{3}{4} + \sqrt{10}\) is irrational. In summary, the sum of a rational number \(\frac{3}{4}\) and an irrational number \(\sqrt{10}\) yields an irrational number because the rational component cannot provide a fractional representation for the inherently non-repeating and non-terminating nature of the irrational number.

Answer 6

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)

Answer 7

To prove that the product of \(2\pi\) and \(\frac{3}{4}\) is an irrational number, we first calculate the product:

\[ 2\pi \times \frac{3}{4} = \frac{3 \times 2\pi}{4} = \frac{6\pi}{4} = \frac{3\pi}{2} \]

Next, we need to determine if \(\frac{3\pi}{2}\) is rational or irrational.

1. Properties of Numbers:

   • The number \(\pi\) is known to be an irrational number. This means it cannot be expressed as the quotient of two integers.
   • A rational multiplier, \(\frac{3}{4}\), does not change the irrationality of \(\pi\).

2. Reasoning: When a rational number (like \(\frac{3}{4}\)) is multiplied by an irrational number (like \(\pi\)), the product is also irrational. Thus, \(\frac{3\pi}{2}\) remains irrational.

Conclusion: The product \(\frac{3\pi}{2}\) is irrational because it results from multiplying the rational number \(\frac{3}{4}\) by the irrational number \(2\pi\). This multiplication preserves the property of irrationality, confirming that the product is not expressible as a fraction of two integers.