Show that the product of two of the numbers 651000 82001 31

Show that the product of two of the numbers 65^1000 8^2001 + 3^177 , 79^1212 9^2399 + 2^2001 , and 24^4493

5^8192 + 7^1777 is non-negative.

Of these three numbers, atleast two must have the same sign (both positive or both negative), since

there are only two signs.

(It is conceivable that some of them are zero, but we view zero as positive for the purposes of this problem.)

The product of two with the same sign is nonnegative. This was a nonconstructive proof, since we have not identified which product is nonnegative.

(In fact, a computer algebra system will tell us that all three are positive, so all three products are positive.)