Let's determine which of the given expressions does NOT have a rational number equivalent by evaluating each expression.
### Choice F
[tex]\[ F: \quad -(\sqrt{7} - \sqrt{13})^2(\sqrt{7} + \sqrt{13})^3(\sqrt{7} - \sqrt{13}) \][/tex]
After simplifying this complex expression involving square roots, it turns out it simplifies to 216, which is a rational number.
### Choice G
[tex]\[ G: \quad (1 + \sqrt{2})(1 - \sqrt{2}) \][/tex]
This can be simplified as follows:
[tex]\[ G = 1^2 - (\sqrt{2})^2 = 1 - 2 = -1 \][/tex]
Since -1 is a rational number, choice G simplifies to a rational number.
### Choice H
[tex]\[ H: \quad \frac{3 \sqrt{6}}{24} - \frac{\sqrt{6}}{12} \][/tex]
Simplify each term:
[tex]\[ \frac{3 \sqrt{6}}{24} = \frac{\sqrt{6}}{8} \][/tex]
[tex]\[ \frac{\sqrt{6}}{12} \text{ remains as it is} \][/tex]
Combine the terms:
[tex]\[ \frac{\sqrt{6}}{8} - \frac{\sqrt{6}}{12} \][/tex]
Find a common denominator (24):
[tex]\[ \frac{3\sqrt{6}}{24} - \frac{2\sqrt{6}}{24} = \frac{(3\sqrt{6} - 2\sqrt{6})}{24} = \frac{\sqrt{6}}{24} \][/tex]
Since [tex]\(\frac{\sqrt{6}}{24}\)[/tex] still contains an irrational number, it is not rational.
### Choice J
[tex]\[ J: \quad \frac{(2 - \sqrt{5})(2 + \sqrt{5})}{4 - 5} - \frac{3 \pi}{7 \pi} \][/tex]
First, simplify the numerator:
[tex]\[ (2 - \sqrt{5})(2 + \sqrt{5}) = 2^2 - (\sqrt{5})^2 = 4 - 5 = -1 \][/tex]
Now the denominator:
[tex]\[ 4 - 5 = -1 \][/tex]
Therefore,
[tex]\[ \frac{-1}{-1} = 1 \][/tex]
Then simplify the other fraction:
[tex]\[ \frac{3 \pi}{7 \pi} = \frac{3}{7} \][/tex]
Thus,
[tex]\[ 1 - \frac{3}{7} = \frac{7}{7} - \frac{3}{7} = \frac{4}{7} \][/tex]
Since [tex]\(\frac{4}{7}\)[/tex] is rational, this expression simplifies to a rational number.
### Choice K
[tex]\[ K: \quad \frac{3 - \pi}{4} + \frac{8 \pi^2}{32 \pi} \][/tex]
First, simplify each fraction:
[tex]\[ \frac{8 \pi^2}{32 \pi} = \frac{8 \pi}{32} = \frac{\pi}{4} \][/tex]
Combining the fractions:
[tex]\[ \frac{3 - \pi}{4} + \frac{\pi}{4} = \frac{3 - \pi + \pi}{4} = \frac{3}{4} \][/tex]
Since [tex]\(\frac{3}{4}\)[/tex] is rational, this expression simplifies to a rational number.
### Conclusion
By evaluating all the choices, we see that:
- Choices F, G, J, and K simplify to rational numbers.
- Choice H simplifies to [tex]\(\frac{\sqrt{6}}{24}\)[/tex], which is not a rational number.
Thus, the expression that does NOT have a rational number equivalent is:
[tex]\[ \boxed{H} \][/tex]
