Question
Question
According to the Closure Property, which of these sums results in a rational number?(1 point)
Responses
36−−√+221−−−√
Start Root 36 End Root plus Start Root 221 End Root
0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction
0.6¯¯¯¯+910
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction
π+4–√
According to the Closure Property, which of these sums results in a rational number?(1 point)
Responses
36−−√+221−−−√
Start Root 36 End Root plus Start Root 221 End Root
0.643892553…+34
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction
0.6¯¯¯¯+910
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction
π+4–√
Answers
GPT-4o mini
Let's evaluate each of the sums to determine whether they result in a rational number.
1. **\( \sqrt{36} + \sqrt{221} \)**:
- \( \sqrt{36} = 6 \) (a rational number)
- \( \sqrt{221} \) is an irrational number (since 221 is not a perfect square).
- The sum \( 6 + \sqrt{221} \) is therefore irrational.
2. **\( 0.643892553... + \frac{3}{4} \)**:
- \( 0.643892553... \) appears to be a non-repeating, non-terminating decimal (irrational) unless specified otherwise, but just in case, let’s sum it with \( \frac{3}{4} \) (which is rational).
- A non-terminating, non-repeating decimal (irrational) plus a rational number results in an irrational number. Therefore, this sum is likely irrational.
3. **\( 0.6\overline{6} + \frac{9}{10} \)**:
- \( 0.6\overline{6} \) is equivalent to \( \frac{2}{3} \) (a rational number).
- \( \frac{9}{10} \) is also a rational number.
- The sum \( \frac{2}{3} + \frac{9}{10} \) can be expressed as:
\[
\text{Common denominator} = 30 \\
\frac{2}{3} = \frac{20}{30}, \quad \frac{9}{10} = \frac{27}{30} \\
\frac{20}{30} + \frac{27}{30} = \frac{47}{30} \text{ (which is rational)}
\]
- Thus, this sum is rational.
4. **\( \pi + \sqrt{4} \)**:
- \( \sqrt{4} = 2 \) (a rational number).
- \( \pi \) is irrational.
- Therefore, \( \pi + 2 \) is irrational.
From the above evaluations, the only sum that results in a rational number is \( 0.6\overline{6} + \frac{9}{10} \).
1. **\( \sqrt{36} + \sqrt{221} \)**:
- \( \sqrt{36} = 6 \) (a rational number)
- \( \sqrt{221} \) is an irrational number (since 221 is not a perfect square).
- The sum \( 6 + \sqrt{221} \) is therefore irrational.
2. **\( 0.643892553... + \frac{3}{4} \)**:
- \( 0.643892553... \) appears to be a non-repeating, non-terminating decimal (irrational) unless specified otherwise, but just in case, let’s sum it with \( \frac{3}{4} \) (which is rational).
- A non-terminating, non-repeating decimal (irrational) plus a rational number results in an irrational number. Therefore, this sum is likely irrational.
3. **\( 0.6\overline{6} + \frac{9}{10} \)**:
- \( 0.6\overline{6} \) is equivalent to \( \frac{2}{3} \) (a rational number).
- \( \frac{9}{10} \) is also a rational number.
- The sum \( \frac{2}{3} + \frac{9}{10} \) can be expressed as:
\[
\text{Common denominator} = 30 \\
\frac{2}{3} = \frac{20}{30}, \quad \frac{9}{10} = \frac{27}{30} \\
\frac{20}{30} + \frac{27}{30} = \frac{47}{30} \text{ (which is rational)}
\]
- Thus, this sum is rational.
4. **\( \pi + \sqrt{4} \)**:
- \( \sqrt{4} = 2 \) (a rational number).
- \( \pi \) is irrational.
- Therefore, \( \pi + 2 \) is irrational.
From the above evaluations, the only sum that results in a rational number is \( 0.6\overline{6} + \frac{9}{10} \).