Question
Question
According to the Closure Property, which of these sums results in a rational number?(1 point)
Responses
0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction
π+4–√
pi plus Start Root 4 End Root
36−−√+221−−−√
Start Root 36 End Root plus Start Root 221 End Root
0.6¯¯¯¯+910
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction
According to the Closure Property, which of these sums results in a rational number?(1 point)
Responses
0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction
π+4–√
pi plus Start Root 4 End Root
36−−√+221−−−√
Start Root 36 End Root plus Start Root 221 End Root
0.6¯¯¯¯+910
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction
Answers
GPT-4o mini
To determine which of these sums results in a rational number, we will analyze each expression:
1. \( 0.643892553\ldots + 34 \)
- The first term is a non-repeating decimal and is thus irrational. Adding 34 (a rational number) to it will still yield an irrational number.
2. \( \pi + 4 - \sqrt{4} \)
- \( \pi \) is irrational, \( 4 \) is rational, and \( \sqrt{4} = 2 \) is rational. The sum of an irrational number (\(\pi\)) and rational numbers (4 and -2) will still be irrational.
3. \( \sqrt{36} + \sqrt{221} \)
- \( \sqrt{36} = 6 \) (a rational number) while \( \sqrt{221} \) is irrational (since 221 is not a perfect square). The sum of a rational number (6) and an irrational number (\(\sqrt{221}\)) will be irrational.
4. \( 0.6\overline{6} + \frac{9}{10} \)
- The decimal \( 0.6\overline{6} \) represents the rational number \( \frac{2}{3} \). When we add \( \frac{2}{3} \) (rational) and \( \frac{9}{10} \) (rational), the result is rational.
Based on this analysis, the only sum that results in a rational number is:
**\( 0.6\overline{6} + \frac{9}{10} \)**.
1. \( 0.643892553\ldots + 34 \)
- The first term is a non-repeating decimal and is thus irrational. Adding 34 (a rational number) to it will still yield an irrational number.
2. \( \pi + 4 - \sqrt{4} \)
- \( \pi \) is irrational, \( 4 \) is rational, and \( \sqrt{4} = 2 \) is rational. The sum of an irrational number (\(\pi\)) and rational numbers (4 and -2) will still be irrational.
3. \( \sqrt{36} + \sqrt{221} \)
- \( \sqrt{36} = 6 \) (a rational number) while \( \sqrt{221} \) is irrational (since 221 is not a perfect square). The sum of a rational number (6) and an irrational number (\(\sqrt{221}\)) will be irrational.
4. \( 0.6\overline{6} + \frac{9}{10} \)
- The decimal \( 0.6\overline{6} \) represents the rational number \( \frac{2}{3} \). When we add \( \frac{2}{3} \) (rational) and \( \frac{9}{10} \) (rational), the result is rational.
Based on this analysis, the only sum that results in a rational number is:
**\( 0.6\overline{6} + \frac{9}{10} \)**.