Four $2$'s to make $7$ So I'm trying to use four $2$'s to make the number $7$. I can use parenthesis, addition, subtraction, indicies, addition, division, multiplication 
edit: also have the same problem with $9$ but I'll try that myself if someone could help with $7$

I previously posted an answer listing every possible real number obtainable this way (assuming $22$ is allowable by using two $2$'s. In the revision history, one can find a table where I assumed that $22$ was not allowed). $7$ is not on the list, hence cannot be thusly formed, assuming my program works (which it does). As has been noted in comments and is present on the list, $9$ may be formed as
$$9=\left(2+\frac{2}2\right)^2.$$
Without resulting to brute force, one may prove that $7$ cannot be formed by noting that the only operations capable of forming anything but even numbers when given even numbers are division and negative exponents. We can rule out negative exponents, because that would leave us with a non-integer and, to create an integer, we would either add that with another non-integer (which would take up all $4$ operations and would be $2^{-2}+2^{-2}\neq 7$), divide by a non-integer (which similarly gives $\frac{2^{-2}}{2^{-2}}\neq 7$), subtract a non-integer (which gives $2^{-2}-2^{-2}\neq 7$) or multiply by an integer (which, seeing that the only possible non-integers with less than $4$ twos are $2^{-2}$ and $2^{-2-2}=(2+2)^{-2}$, doesn't give any useful forms. The casework is left as an exercise). We can similarly eliminate the possibility of division yielding a non-integer. Thus, we can conclude that division occurs somewhere, yielding an odd number. If $\frac{2}2$ occurs, then our numbers are $2$, $2$, and $1$ where the largest number that can occur without an exponent is $2+2+1<5$, which is too small. So exponentiation must occur, but for this to be helpful, either the base or the exponent must be at least $3$, giving that $2^3$ and $3^2$ are the only integers numbers larger than $5$ obtainable if we include $\frac{2}2$. Thus, $\frac{2}2$ may not occur either. One may find that the only other form yielding an odd number is $\frac{2+2}{2+2}$ (or many other expressions evaluating to $\frac{4}4$). This is, more or less, a proof.
