Astronauts found the following symbol sequence on an alien planet:

$$
\%!\&+!0++!!!
$$

They somehow figured out that an inhabitant of the planet performed some operation with natural numbers here (addition, subtraction, multiplication, division, exponentiation, or root extraction). They know that the planet uses some positional number system, but they do not know the base of the number system, the direction in which consecutive digits are written, or how the operations are denoted. We assume that both the operation symbol and the equality symbol are each represented by one symbol. What could the calculation have been?
Since we have assumed that the operational symbol and the equality symbol are each represented by a single symbol, these can only be chosen from the %, &, and 0 symbols, as the + and ! symbols occur more than once, and next to each other.

The %-symbol cannot be the equality symbol due to its placement. It could only be an operational symbol if it represented an operation performed on a "single number," otherwise, a separator symbol would be needed between the numbers involved in the operation, and the operation would not be represented by a single symbol. By "operation on a single number," we specifically mean that the %-symbol represents a square root; the base 2 is not written, similar to Earth conventions. However, this idea does not hold, as the result of the square root would be a multi-digit number, larger than the number from which the root was taken. Therefore, the %-symbol represents a digit.

First, let's consider the possibility that the equality symbol is the 0, and the & is the operational symbol. In this case, the two numbers are \% ! and + !, and the result of the operation is $++!$ ! !.

The operation cannot be addition, subtraction, multiplication, or division, as the result is a five-digit number, and the operation is performed with two two-digit numbers. (We assume that in the aliens' writing system, a zero digit cannot be "in front," or that they use as many symbols to write a number as the number of digits it has.) We can also rule out the square root. The operation can only be exponentiation.

Since three different digits appear, the base of the number system must be at least 3.

Let's first examine the assumption that the base of the number system used is 3. In this case, the number on the right side of the equality cannot be greater than $\left(3^{5}-1\right)$. Since the base of the power on the left side is at least 3, the exponent can be at most 4. However, in the base-3 number system, 4 must be written with two identical digits, so the exponent can only be 3.

If \% ! is the base and + ! is the exponent, then with the same writing direction as ours, the values of the symbols are: $+=1,!=0, \%=2$. The operation in the base-3 system: $20^{10}=11000$, in the decimal system: $6^{3}=108$. The equality is not correct.

If the writing direction is opposite to ours, then the values of the symbols are: $!=1,+=0, \%=2$. The operation in the base-3 system: $12^{10}=11100$, in the decimal system: $5^{3}=125$.

The equality is not correct.

If + ! is the base and \% ! is the exponent, then with the same writing direction as ours, the values of the symbols are: $\%=1,!=0,+=2$. The operation in the base-3 system: $20^{10}=22000$, in the decimal system: $6^{3}=216$. The equality is correct. If the writing direction is opposite to ours, the values of the symbols are: $!=1, \%=0,+=2$. The operation in the base-3 system: $12^{10}=11122$, in the decimal system: $5^{3}=125$. Again, we get a correct equality.

Now let's extend our examination to all number systems with a base greater than 3. If the base of the number system is $z$, then the smallest power that can be produced with a two-digit exponent is $z^{z}$, while the largest five-digit number is $\left(z^{5}-1\right)$. Therefore, $z$ must be less than 5. If the base of the number system used is 4, the exponent in the operation cannot be greater than 4, but it cannot be smaller either, as it is a two-digit number. So the exponent is 4. In this case, the base must be at least 6, as 5 would be written with two identical symbols, and 4 would be the same two symbols as the exponent. However, $6^{4}>4^{5}-1$, so the base-4 number system is also not possible.

We must still consider the possibility that & is the equality symbol, and 0 is the operational symbol. That is, the result is \% !, and the two operands are + ! and $++!!$.

In this case, it could be the inverse of an exponentiation, the root extraction, for which we previously obtained a correct equality. However, we only got a correct equality when + ! was the base and \% ! was the exponent, which would result in a situation where the equality symbol separates the root index from the root extraction symbol and the "under the root" quantity. This operation is the other inverse of exponentiation: the logarithm (we know the base and the result, we are looking for the exponent). Although this possibility did not arise when the problem was formulated, we cannot definitively say that the inhabitants of the alien planet did not have a special symbol for this, just as we do for addition.

Summarizing our results: the alien used the base-3 number system and performed either an exponentiation, calculating $5^{3}$ or $6^{3}$, or a logarithm, calculating the logarithm of 125 to base 5 or the logarithm of 216 to base 6.