If 𝑟 is equal to two plus seven 𝑖 and 𝑠 is equal to one plus two 𝑖, what is 𝑟 multiplied by 𝑠?
In this question, we’re given two complex numbers and we’re asked to find the product of these two complex numbers. So the first thing we can do is write this product out in full. We have 𝑟 multiplied by 𝑠 is equal to two plus seven 𝑖 multiplied by one plus two 𝑖.
And now we can see something interesting. Inside of each of our parentheses, we have two terms. And we know how to distribute parentheses in this form by using the FOIL method. The FOIL method tells us we need to start by multiplying the first two terms in our parentheses. This gives us two multiplied by one. Next, the FOIL method tells us we need to add on the products of our two outer terms. And we can see that’s two multiplied by two 𝑖. The third step in the FOIL method is to add on the product of our two inner terms. And in this product, that’s seven 𝑖 multiplied by one. And finally, the last step of the FOIL method is to add on the product of our two last terms. So in this case, we need to add on seven 𝑖 multiplied by two 𝑖.
So by using the FOIL method, we’ve distributed our parentheses and now we’ve written it as four terms. We can now simplify each of these terms separately. First, two multiplied by one is equal to two. Next, to multiply two and two 𝑖 together, we need to multiply two by two to get four. And then we multiply this by 𝑖, giving us four 𝑖.
Next, any number multiplied by one is equal to itself. So seven 𝑖 times one is just equal to seven 𝑖. And simplifying our last term is more complicated. We need to multiply seven by two, and this gives us 14. However, this term still has two factors of 𝑖. So we need to multiply this by 𝑖 times 𝑖, which we’ll write as 𝑖 squared.
So, so far, we’ve shown that 𝑟 multiplied by 𝑠 is equal to two plus four 𝑖 plus seven 𝑖 plus 14𝑖 squared. And we can simplify this even further. We can add our second and third term together. To do this, we just add their coefficients of 𝑖 together. Four 𝑖 plus seven 𝑖 is equal to 11𝑖. And there’s still more simplification we can do.
We can notice in our third and final term we have a factor of 𝑖 squared. And 𝑖 is the square root of negative one. So 𝑖 squared is the square root of negative one all squared, which must be equal to negative one. So in this expression, we can replace 𝑖 squared with negative one, which means instead of adding 14 𝑖 squared we’re now subtracting 14. And of course this lets us simplify our expression since we can simplify two minus 14. We can calculate this is equal to negative 12, giving us our final answer that 𝑟 multiplied by 𝑠 is equal to negative 12 plus 11𝑖.