This puzzle is inspired by JLee's What is a Word/Phrase™ series and the subsequent "Number" variants. (Actually, I'd originally tried to create a more original puzzle using the same idea, but it turned into another one of these. Sigh. Maybe the types of numbers given lend this somewhat more novelty.)
If a positive real number obeys a special rule, I call it a Good Pasta Number™. Here are some examples:
| A good, solid pasta! Yeah! | You are a horrible cook |
|---|---|
| $6\sqrt{6}$ | $3\sqrt{5}$ |
| $6$ | $8$ |
| $3\sqrt{6}$ | $6\sqrt{3}$ |
| $\frac{3\sqrt{2}+\sqrt{6}}{2}$ | $\frac{3\sqrt{6}+\sqrt{2}}{2}$ |
| $3\sqrt{50-22\sqrt{5}}$ | $3\sqrt{50+22\sqrt{5}}$ |
| $9\sqrt{3}-3\sqrt{15}$ | $18\sqrt{3}-6\sqrt{15}$ |
| $\frac{42\sqrt{6}}{23}$ | $\frac{56\sqrt{6}}{19}$ |
| $\frac{15\sqrt{6}-18\sqrt{3}+135\sqrt{2}-162}{14}$ | $\frac{15\sqrt{30}-18\sqrt{15}+135\sqrt{10}+30\sqrt{6}-162\sqrt{5}-36\sqrt{3}+270\sqrt{2}-384}{14}$ |
There are many more Good Pasta Numbers™ not shown above, but only a finite amount.
Which numbers make a good pasta™?
Note:
There's nothing hidden in the flavortext other than a very subtle connection (not really a clue); this really is just a normal "What is a _______" puzzle. I like making my puzzles fun to read. Also, the property is specific to the numbers themselves, and not any specific representation of the aforementioned numbers.
