Question

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

0.643892553...+34

√36+√221

π+√4

0.6¯+9/10

Answer 1

To determine which sums result in a rational number based on the Closure Property of rational numbers, we need to analyze each option:

1. **0.643892553... + 34**:
The number 34 is a rational number, and 0.643892553... appears to be a decimal representation of a rational number (it does not terminate, but whether it's repeating or not is unclear). However, regardless of its behavior, if we add a rational number (34) to another rational number (0.643892553...), the result will be rational. Therefore, this sum is rational.

2. **√36 + √221**:
√36 = 6, which is rational, but √221 is not a perfect square, so it is an irrational number. The sum of a rational number (6) and an irrational number (√221) is irrational. Thus, this sum is not a rational number.

3. **π + √4**:
√4 = 2, which is rational, but π is an irrational number. The sum of rational (2) and irrational (π) numbers is irrational. Thus, this sum is not a rational number.

4. **0.6̅ + 9/10**:
The number 0.6̅ (repeating 6) is equivalent to 2/3, which is rational. The fraction 9/10 is also rational. The sum of two rational numbers (2/3 and 9/10) will be rational. Therefore, this sum is rational.

In summary, the sums that result in a rational number are:

1. **0.643892553... + 34** (if 0.643892553... is rational)
4. **0.6̅ + 9/10**

Thus, the valid sums that yield a rational result are the first and fourth options.