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

Question

Responses

A) π+4–√
pi plus Start Root 4 End Root

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

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

D) 0.643892553…+3/4
0.643892553 dot dot dot plus Start Fraction 3 over 4 End Fraction

Answer 1

To determine which of the sums results in a rational number, we can analyze each option based on the Closure Property of rational numbers, which states that the sum (or product) of two rational numbers is rational, while the sum (or product) involving irrational numbers may or may not be rational.

A) \( \pi + \sqrt{4} \)

• \( \pi \) is irrational, and \( \sqrt{4} = 2 \) is rational. The sum of a rational and an irrational number is irrational.
• Not a rational number.

B) \( 0.6\bar{6} + \frac{9}{10} \)

• \( 0.6\bar{6} \) (which is \( \frac{2}{3} \)) is rational, and \( \frac{9}{10} \) is also rational. The sum of two rational numbers is rational.
• This is a rational number.

C) \( \sqrt{36} + \sqrt{221} \)

• \( \sqrt{36} = 6 \) is rational, but \( \sqrt{221} \) is irrational. The sum of a rational and an irrational number is irrational.
• Not a rational number.

D) \( 0.643892553\ldots + \frac{3}{4} \)

• The number \( 0.643892553\ldots \) appears to be a non-repeating, non-terminating decimal, suggesting it is irrational. \( \frac{3}{4} \) is rational. The sum of a rational and an irrational number is irrational.
• Not a rational number.

Thus, the only sum that results in a rational number is B) \( 0.6\bar{6} + \frac{9}{10} \).