Simplify seven plus six 𝑖 multiplied by two minus nine 𝑖.
In this question, we were asked to simplify the product of two complex numbers given in algebraic form. And we can notice something interesting. This is exactly the same as multiplying the product of two binomials. A complex number given in algebraic form has two terms, and a binomial also has two terms. So we can distribute the parentheses by using the FOIL method. Recall in the FOIL method, we start by multiplying the first two terms in our parentheses. And we can see this is seven multiplied by two. Next, the FOIL method tells us to multiply the two outer terms together. This means we need to add on seven multiplied by negative nine 𝑖.
After this, we need to add on the products of our two inner terms; in this case, that’s six 𝑖 multiplied by two. Finally, the FOIL method tells us we need to add on the product of the last two terms. And in this case, that’s six 𝑖 multiplied by negative nine 𝑖. And now we can start simplifying. First, seven multiplied by two is equal to 14. Next, seven multiplied by negative nine is equal to negative 63. So seven multiplied by negative nine 𝑖 is negative 63𝑖. Similarly, to simplify six 𝑖 multiplied by two, we need six multiplied by two is equal to 12. This is just 12𝑖. And in our last term, we have six 𝑖 multiplied by negative nine 𝑖. Well, six multiplied by negative nine is equal to negative 54. But then we still need to have 𝑖 multiplied by 𝑖 which is equal to 𝑖 squared.
So this gives us 14 minus 63𝑖 plus 12𝑖 minus 54𝑖 squared. And to simplify this, we need to recall what we mean by 𝑖. 𝑖 is the square root of negative one. So 𝑖 squared is going to be equal to negative one. Therefore, we can use this to simplify the last term in our expression. Since 𝑖 squared is negative one, we’re now subtracting negative 54. And of course, subtracting negative 54 is the same as adding 54. So we can rearrange this expression to give us 14 plus 54 plus 12𝑖 minus 63𝑖. Then all we need to do is compute like terms. 14 plus 54 is equal to 68, and 12𝑖 minus 63𝑖 is negative 51𝑖, giving us our final answer of 68 minus 51𝑖. Therefore, we were able to show by using the FOIL method, seven plus six 𝑖 multiplied by two minus nine 𝑖 is equal to 68 minus 51𝑖.