Question

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

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

0.6¯¯¯¯+910

Answer 1

The Closure Property states that when you perform an operation (like addition) on two numbers of a certain type (like rational numbers), the result will also be of that type. Rational numbers are numbers that can be expressed as the quotient of two integers, where the denominator is not zero.

Let's analyze each option:

1. **π + √4**:
- Here, π (pi) is an irrational number, and √4 equals 2, which is rational. The sum of a rational number and an irrational number is irrational.
- Result: **Irrational**

2. **0.643892553... + ¾**:
- The number 0.643892553... is a rational number (since it can be expressed as a fraction), and ¾ is clearly rational as well. The sum of two rational numbers is rational.
- Result: **Rational**

3. **√36 + √221**:
- √36 is equal to 6, which is rational, while √221 is irrational (it cannot be expressed as a fraction). The sum of a rational and an irrational number is irrational.
- Result: **Irrational**

4. **0.6̅ + 910**:
- The number 0.6̅ (which represents a repeating decimal, equivalent to 2/3) is rational. The number 910 is also rational (it can be expressed as 910/1). The sum of two rational numbers is rational.
- Result: **Rational**

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

- **0.643892553... + ¾**
- **0.6̅ + 910**

Thus, the correct responses are the second and fourth options.