According to the Closure Property, which of these differences would result in a rational number?(1 point)

Question

Responses

64−−√−7–√
square root of 64 minus square root of 7

49−−√−14
square root of 49 minus 1 fourth

5–√−3.2
square root of 5 minus 3 point 2

6.2¯¯¯−π

Answer 1

To determine which of these differences would result in a rational number, we need to evaluate each option one by one:

1. \(\sqrt{64} - \sqrt{7}\):

   • \(\sqrt{64} = 8\) (rational)
   • \(\sqrt{7}\) is irrational.
   • Thus, \(8 - \sqrt{7}\) is irrational.

2. \(\sqrt{49} - \frac{1}{4}\):

   • \(\sqrt{49} = 7\) (rational)
   • \(\frac{1}{4}\) is also rational.
   • Thus, \(7 - \frac{1}{4} = 7 - 0.25 = 6.75\) (rational).

3. \(\sqrt{5} - 3.2\):

   • \(\sqrt{5}\) is irrational.
   • \(3.2\) is rational.
   • Thus, \(\sqrt{5} - 3.2\) is irrational.

4. \(6.2\overline{6} - \pi\):

   • \(6.2\overline{6}\) is a rational number (specifically, it is equal to \(6 \frac{2}{3}\)).
   • \(\pi\) is irrational.
   • Thus, \(6.2\overline{6} - \pi\) is irrational.

The only difference that results in a rational number is:

\(\sqrt{49} - \frac{1}{4}\), which equals \(6.75\).