According to the Closure Property, which of these sums results in a rational number?(1 point)

Question

Responses

π+4–√
pi plus Start Root 4 End Root

0.6¯¯¯¯+910
0 point Modifying above 6 with bar plus Start Fraction 9 over 10 End Fraction

36−−√+221−−−√
Start Root 36 End Root plus Start Root 221 End Root

0.643892553…+34

Answer 1

To determine which of the given sums results in a rational number, we need to analyze each option based on the properties of rational and irrational numbers.

1. π + √4

   • \( \pi \) is an irrational number and \( \sqrt{4} = 2 \) is a rational number. The sum of an irrational number and a rational number is irrational. Therefore, this option does not yield a rational number.

2. 0.6̅ + \frac{9}{10}

   • \( 0.6̅ \) (which represents \( 0.6666\ldots \)) is a rational number since it can be expressed as \( \frac{2}{3} \).
   • \( \frac{9}{10} \) is clearly a rational number as well.
   • The sum of two rational numbers is rational. Therefore, \( 0.6̅ + \frac{9}{10} \) results in a rational number.

3. √36 + √221

   • \( \sqrt{36} = 6 \), which is rational.
   • \( \sqrt{221} \) is an irrational number (since 221 is not a perfect square). The sum of a rational number and an irrational number is irrational. Thus, this option does not yield a rational number.

4. 0.643892553… + 34

   • \( 0.643892553\ldots \) is a non-repeating decimal, which makes it an irrational number.
   • \( 34 \) is a rational number. The sum of an irrational number and a rational number is again irrational. Therefore, this option does not yield a rational number.

Based on the analysis, the only sum that results in a rational number is:

0.6̅ + \frac{9}{10}.