main.cpp

#include <iostream>
#include <string>
#include <vector>

//method: parse the string and create a vector of numbers and operators then iterate that vector to create the value
int main(){
    std::string str_val = "3*5+11+5*20+9*2*4+135";
    std::vector<std::string> expression;
    std::string number_holder;
    int position = 0;
    for (std::string::iterator it=str_val.begin(); it!=str_val.end(); ++it) {
        std::cout << "position: " << position << std::endl;
        position++;

        if(std::isdigit(*it)){
            number_holder.push_back(*it);
            std::cout << "number is: " << number_holder << std::endl;
        }
        else{
            //if the next char is not a number then its an operator

            expression.push_back(number_holder);
            number_holder = "";

            std::cout << "operator is: " << *it << std::endl;
            std::string str(1, *it);
            expression.push_back(str);
        }
    }

    // There will always be a number as the last operand, so lets add it to the expression vector
    std::cout << "number is: " << number_holder << std::endl;
    expression.push_back(number_holder);

    // lets check the expression vector! TODO: delete this code segment, only for debuging


    //Great! Now lets while loop over the expression vector until its size == 1.
    int value = 0;
    //expression.size()>1
    bool complete_multiplication = false;
    while(!complete_multiplication){
        //view current expression
        std::cout << "expression" << std::endl;
        for(std::vector<std::string>::iterator debug_it=expression.begin(); debug_it!=expression.end(); ++debug_it){
            std::cout << *debug_it << std::endl;
        }
        for (unsigned i=0; i<expression.size(); ++i){
            if (expression[i] == "*"){
                expression[i-1] = std::to_string(std::stoi(expression[i-1])*std::stoi(expression[i+1]));
                expression.erase(expression.begin()+i);
                expression.erase(expression.begin()+i);
                break;
            }
            if (i==expression.size()-1) {
                complete_multiplication = true;
                std::cout << "complete multiplication" << std::endl;
            }

        }

        std::cout << "value: " << value << std::endl;
    }
    bool complete_addition = false;
    while(!complete_addition){
        //view current expression
        std::cout << "expression" << std::endl;
        for(std::vector<std::string>::iterator debug_it=expression.begin(); debug_it!=expression.end(); ++debug_it){
            std::cout << *debug_it << std::endl;
        }
        for (unsigned i=0; i<expression.size(); ++i){
            if (expression[i] == "+"){
                expression[i-1] = std::to_string(std::stoi(expression[i-1])+std::stoi(expression[i+1]));
                expression.erase(expression.begin()+i);
                expression.erase(expression.begin()+i);
                break;
            }
            if (i==expression.size()-1) {
                complete_addition = true;
                std::cout << "complete addition" << std::endl;
            }

        }

    }

    return 0;
}

not pretty but here is the console output. scroll down to the bottom and you can see the only value left in the expression vector is the answer 333.

position: 0
number is: 3
position: 1
operator is: *
position: 2
number is: 5
position: 3
operator is: +
position: 4
number is: 1
position: 5
number is: 11
position: 6
operator is: +
position: 7
number is: 5
position: 8
operator is: *
position: 9
number is: 2
position: 10
number is: 20
position: 11
operator is: +
position: 12
number is: 9
position: 13
operator is: *
position: 14
number is: 2
position: 15
operator is: *
position: 16
number is: 4
position: 17
operator is: +
position: 18
number is: 1
position: 19
number is: 13
position: 20
number is: 135
number is: 135
expression
3
*
5
+
11
+
5
*
20
+
9
*
2
*
4
+
135
value: 0
expression
15
+
11
+
5
*
20
+
9
*
2
*
4
+
135
value: 0
expression
15
+
11
+
100
+
9
*
2
*
4
+
135
value: 0
expression
15
+
11
+
100
+
18
*
4
+
135
value: 0
expression
15
+
11
+
100
+
72
+
135
complete multiplication
value: 0
expression
15
+
11
+
100
+
72
+
135
expression
26
+
100
+
72
+
135
expression
126
+
72
+
135
expression
198
+
135
expression
333
complete addition

Process finished with exit code 0