C/C++ :: Binomial Expansion - Simple Math String Into Array

My project is binomial expansion. I want to allow for basic string input. Then, I must process the string for input into the other portions of my program. To make the binomial expansion calculation simpler I am using two separate arrays for the first and second term of the binomial expansion. The integer array structure for each term is [{numerical Coefficient of term}, {exponent of variable a}, {exponent of variable b}, {exponent of variable c}, ......{exponent of variable x}, {exponent of variable y}, exponent of variable z}].

So, if the user doesn't use a variable it's exponent's value will = 0. The reason why I set my array up like this is so it would be very simple to raise each term to the power of n, or (n-k) (Multiple variable exponents by number, raise #coefficient to the power). It would also be easy to combine the terms to get one term. terms[0] are multiplied, all other terms are added.

My issue is translating a basic mathematical string consisting of only Integers, letter, parenthesis, and '^', into the array. I find difficulty mainly due to the '^' symbol. I know how to find the array index for a given letter. termArrayIndex = (int)((char from string) - 'a' + 1)

the main things causing me issues

-The issue is knowing which numbers to multiply by the array.

E.g. (_(_____)^#(__^#__)^#____)^#

I started by finding the first ')', then finding the preceding '(' and processing that string. I would then remove said string and use recusrion to go through the whole string. This method doesn't work because there is data loss upon removal of processed string. EG (__(__)^#) =remove innermost string and parenthesis=> (__^#) != (__()^2,

-maybe enter all terms, leave parenthesis, then process exponents separately...

-I may be able to find the # of separate terms by seeing # of times a '(' comes after a ')'

This is by far the most difficult part of the problem for me. But that also makes it the most fun. I want it to be able to handle all variation of these basic mathematical expressions.