According to the Closure Property, which of these sums would result in a rational number?(1 point) Responses 12−−√+14 square root of 12 plus 1 fourth 5–√+7.89 square root of 5 plus 7 point 8 9 8.5¯¯¯+π 8 point repeating 5 plus pi 49−−√+2.33

Answer 1

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

A rational number is a number that can be expressed as the quotient of two integers. An irrational number is a number that cannot be expressed as a quotient of two integers.

Let's examine each sum:

1. \(\sqrt{12} + \frac{1}{4}\):

   • \(\sqrt{12}\) is an irrational number (since it cannot be expressed as a fraction).
   • \(\frac{1}{4}\) is a rational number.
   • The sum of an irrational number and a rational number is irrational.
   • Result: Irrational

2. \(\sqrt{5} + 7.89\):

   • \(\sqrt{5}\) is an irrational number.
   • \(7.89\) is a rational number.
   • The sum of an irrational number and a rational number is irrational.
   • Result: Irrational

3. \(8.5 + \pi\):

   • \(8.5\) is a rational number (it can be expressed as \( \frac{17}{2} \)).
   • \(\pi\) is an irrational number.
   • The sum of a rational number and an irrational number is irrational.
   • Result: Irrational

4. \(\sqrt{49} + 2.33\):

   • \(\sqrt{49} = 7\) is a rational number.
   • \(2.33\) is also a rational number.
   • The sum of two rational numbers is rational.
   • Result: Rational

Therefore, the only sum that results in a rational number is:

\(\sqrt{49} + 2.33\) (which equals 7 + 2.33 = 9.33).