Question

According to the Closure Property, which of these sums results in a rational number?(1 point) Responses 0.6¯¯¯¯+910 0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction π+4–√ pi plus Start Root 4 End Root 0.643892553…+34 0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction 36−−√+221−−−√

Answers

Answered by GPT-4o mini
To determine which of the sums results in a rational number, let's analyze each option:

1. **\(0.6\overline{6} + \frac{9}{10}\)**

\(0.6\overline{6}\) is a repeating decimal which is equal to \(\frac{2}{3}\). So, we have:
\[
\frac{2}{3} + \frac{9}{10}
\]
To add these, we need a common denominator. The least common multiple of 3 and 10 is 30:
\[
\frac{2}{3} = \frac{20}{30}, \quad \frac{9}{10} = \frac{27}{30}
\]
Now add:
\[
\frac{20}{30} + \frac{27}{30} = \frac{47}{30}
\]
This is a rational number.

2. **\(\pi + 4 - \sqrt{\pi} + \sqrt{4}\)**

Since \(\pi\) is irrational, and \(\sqrt{\pi}\) is also irrational, we cannot have a sum of these terms resulting in a rational number. \(4\) is rational and \(\sqrt{4} = 2\) is rational, but the sum involving \(\pi\) and \(\sqrt{\pi}\) will remain irrational.

3. **\(0.643892553... + \frac{3}{4}\)**

The decimal \(0.643892553...\) appears to be a non-terminating, non-repeating decimal, which is irrational. Therefore, adding this to a rational number \(\frac{3}{4}\) will also result in an irrational number.

4. **\(\sqrt{36} - \sqrt{221}\)**

\(\sqrt{36} = 6\), which is rational, but \(\sqrt{221}\) is irrational (because 221 is not a perfect square). Thus, \(6 - \sqrt{221}\) results in an irrational number.

In conclusion, the only sum that results in a rational number is:
**\(0.6\overline{6} + \frac{9}{10}\) which equals \(\frac{47}{30}\).**

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