i know everyone plays with their own rules, but from my perspective, a square root spends a 2 - it's not an operator like + or ! . there's an implicit "2" since you could also take the cube root or fourth root
Multiplication is adding numbers to themselves. Exponentiation is multiplying numbers by themselves. Ditto division/taking roots. Squaring a number is just the instance of exponentiation that happens to have two operands. If taking a square root is an inherent use of a two, then dividing a number by another number should also be a use of a two, because there are two operands.
Alternatively, one can claim sqrt is a single symbol, and there are extensions requiring a digit. Log, ln, also have hidden bases, but they’re also valid symbols. Cos/sin may be in radians or degrees, both in common use, and no real place to denote it.
One could even claim division is raising to the -1 power, thereby disqualifying it in exactly the same manner you disqualify sqrt. I’ve never seen sqrt with a 2 index; power -1 is seen quite often for division.
there is a related problem (inspired by an interview questions from ITA software, a part of Google now) that asks to find the smallest positive integer that cannot be expressed using a digit K (in [2,9]) N times in an expression that involves the four basic arithmetic operations and the parentheses: +, -, *, /, (, and ).
to make the problem more interesting, the list of operations can be extended to include exponentiation and concatenation.
for example, using the four basic operations for N=4, and K in [2,9], we get:
7 cannot be computed in 4 2s;
11 cannot be computed in 4 3s;
10 cannot be computed in 4 4s;
8 cannot be computed in 4 5s;
9 cannot be computed in 4 6s;
4 cannot be computed in 4 7s;
5 cannot be computed in 4 8s;
4 cannot be computed in 4 9s;
Squaring is to multiplication as is the latter to addition. If roots are invalid then so should be multiplications.
2x3= 2+2+2 AxB= A+A B times.
Raising to a power is repeated multiplication.
2^3= 2x2x2 A^B = AxA B times
Roots are the inverse of powers
Division is the inverse of multiplications.
Given this… the OC’s comment about giving exceptions to powers in the rules is nonsense because they are still operations.
One could even claim division is raising to the -1 power, thereby disqualifying it in exactly the same manner you disqualify sqrt. I’ve never seen sqrt with a 2 index; power -1 is seen quite often for division.
Go figure….
to make the problem more interesting, the list of operations can be extended to include exponentiation and concatenation.
for example, using the four basic operations for N=4, and K in [2,9], we get:
7 cannot be computed in 4 2s; 11 cannot be computed in 4 3s; 10 cannot be computed in 4 4s; 8 cannot be computed in 4 5s; 9 cannot be computed in 4 6s; 4 cannot be computed in 4 7s; 5 cannot be computed in 4 8s; 4 cannot be computed in 4 9s;
And JFYI since December is near: https://rant.gulbrandsen.priv.no/hack/advent-calendar-for-ne...
7 = ⌈ 2.2² + 2 ⌉