>>>It seems like only perfect squares and multiples of perfect
>>>squares (4*49) have obviously rational square roots
>
>>That's correct. A square root of a non-perfect square
>>(integer or rational) is irrational.
>
>At least it seems that way. It also seems that any number
>of perfect squares multiplied together will create a perfect
>square.

Of course.

>Seems like a direct corollary to the prime permutations of
>the Fundamental Theorem of Arithmetic.

It is simpler

a²×b² = (ab)²

>I wonder if there are subtle distinctions between prime
>perfect squares and compound perfect squares...

Do not know about subtle. But "prime perfect squares" have only one factor different from 1 and the number itself; others have more than 1 such factors.

>>>Is there a way to tell the difference between a true irrational
>>>number and a huge rational number?

No. There is no way to tell one from another by looking at a finite part of there decimal expansion.

>I know it's difficult to judge by expansions.

It is not just difficult, it is simply impossible.

>If you get enough digits, OK.

No, there is never enough digits. It's like asking for the largest number. Give me as long an expansion as you wish and I will modify something beyond what you see to spoil possible rationality of the number.

>I was just wondering
>if there was some mathematical (vs. manual) method of
>working out whether a given decimal expansion was
>rational, irrespective of the length of its period.

Again, no. You need the whole thing.

>Has the "a square root of a non-perfect square is
>irrational" statement been proven, somewhere?

Yes. See Proof 5 at

http://www.cut-the-knot.org/proofs/sq_root.shtml

>
>>>And finally, of all those irrational-looking decimal numbers,
>>>which is the first one (not a perfect square) to be
>>>deceptively rational?
>
>>I do not understand that question but suspect that the
>>answer is there is no first in whatever sense.
>
>If I am looking at a bunch of non-integer results from
>processing square roots, and I do not know of the "rule"
>that says the non-perfect-square roots are irrational,
>I might wonder, which, if any of them, are rational.

But now that you know the rule the mystery is gone, right?

>Especially after all the "hype" about
>2^.5 (square root of 2) being ***irrational***!!! Well, it
>looks like there's a LOT of irrational sqrt()s, so what
>could the big deal be about sqrt(2)?

sqrt(2) is the diagonal of the square with side 1.

>I'm wondering if, among all the *seemingly* irrational
>results,
>there might be a rational result (like 7.142857142857), and
>what
>(among the irrational masses) the first (smallest) one might
>be.

There is no first as there is no last.