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)

Answer 1

To prove that the sum of \(34\) and \(\sqrt{10}\) is irrational, we first note the properties of rational and irrational numbers.

1. Definition of Rational and Irrational Numbers:

   • A number is called rational if it can be expressed as the quotient of two integers, i.e., \( \frac{a}{b} \) where \( a \) and \( b \) are integers and \( b \neq 0\).
   • A number is called irrational if it cannot be expressed in this form.

2. Characteristics of the Given Numbers:

   • The number \(34\) is a rational number because it can be expressed as \( \frac{34}{1} \).
   • The number \(\sqrt{10}\) is an irrational number because 10 is not a perfect square, meaning there are no integer solutions to the equation \(x^2 = 10\).

3. Sum of a Rational and an Irrational Number:

   • The sum of a rational number (in this case, \(34\)) and an irrational number (in this case, \(\sqrt{10}\)) is always irrational.

4. Conclusion:

   • Therefore, \(34 + \sqrt{10}\) must be irrational.

Final Statement: The sum \(34 + \sqrt{10}\) is irrational because it is the sum of a rational number and an irrational number, and the sum of a rational and an irrational number is always irrational.

Answer 2

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)

Answer 3

To prove that the product of \(2\pi\) and \(34\) is an irrational number, we begin by analyzing both factors.

1. Identify the Factors:

   • The number \(34\) is a rational number, as it can be expressed as \( \frac{34}{1} \).
   • The number \(2\pi\) is an irrational number because \(\pi\) is known to be irrational, and any non-zero rational multiple of an irrational number is also irrational.

2. Product of a Rational and an Irrational Number:

   • The product of a rational number (here \(34\)) and an irrational number (here \(2\pi\)) is always irrational.

3. Calculate the Product: \[ 34 \times 2\pi = 68\pi \]

4. Conclusion:

   • Thus, the product \(68\pi\) is irrational.

Final Statement: The product \(68\pi\) is irrational because it is the product of a non-zero rational number (\(34\)) and an irrational number (\(2\pi\)), resulting in an irrational number.