Question
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)
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Answers
GPT-4o mini
To prove that the sum \( 34 + \sqrt{10} \) is irrational, we start by noting two key facts:
1. **Rational and Irrational Numbers**: A rational number can be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \). An irrational number cannot be expressed as such a fraction.
2. **Sum of Rational and Irrational Numbers**: The sum of a rational number and an irrational number is always irrational.
Now, let's break down the components of our expression \( 34 + \sqrt{10} \):
- The number \( 34 \) is a rational number since it can be expressed as \( \frac{34}{1} \).
- The number \( \sqrt{10} \) is known to be irrational (its decimal expansion continues without repeating).
Since \( 34 \) is rational and \( \sqrt{10} \) is irrational, we can apply the principle mentioned above:
\[
34 + \sqrt{10} \text{ (rational)} + \text{ (irrational)} \implies \text{ (irrational)}
\]
Thus, \( 34 + \sqrt{10} \) is irrational.
In summary, \( 34 + \sqrt{10} \) is irrational because it is the sum of a rational number (34) and an irrational number (\(\sqrt{10}\)), which results in an irrational number.
1. **Rational and Irrational Numbers**: A rational number can be expressed as a fraction \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \). An irrational number cannot be expressed as such a fraction.
2. **Sum of Rational and Irrational Numbers**: The sum of a rational number and an irrational number is always irrational.
Now, let's break down the components of our expression \( 34 + \sqrt{10} \):
- The number \( 34 \) is a rational number since it can be expressed as \( \frac{34}{1} \).
- The number \( \sqrt{10} \) is known to be irrational (its decimal expansion continues without repeating).
Since \( 34 \) is rational and \( \sqrt{10} \) is irrational, we can apply the principle mentioned above:
\[
34 + \sqrt{10} \text{ (rational)} + \text{ (irrational)} \implies \text{ (irrational)}
\]
Thus, \( 34 + \sqrt{10} \) is irrational.
In summary, \( 34 + \sqrt{10} \) is irrational because it is the sum of a rational number (34) and an irrational number (\(\sqrt{10}\)), which results in an irrational number.