Express five plus five 𝑖 times eight minus four 𝑖 over one minus 𝑖 in the form 𝑎 plus 𝑏𝑖.
In this question, we’re dividing the product of two complex numbers by another complex number. And so we recall how we divide by complex numbers. We write the quotient in fraction form as in this question. And then we multiply both the numerator and denominator of our fraction by the conjugate of the denominator. And when we talk about the conjugate, we’re essentially changing the sign of the imaginary part of our complex number. So if we have the general form 𝑧 as being 𝑎 plus 𝑏𝑖, the conjugate which we denote as 𝑧 bar is 𝑎 minus 𝑏𝑖. So if we take the quotient here, the conjugate of one minus 𝑖 is one plus 𝑖. So we’re going to multiply both the numerator and denominator of our fraction by one plus 𝑖.
Let’s begin by calculating the denominator. That’s one minus 𝑖 times one plus 𝑖. We multiply the first number in each expression. One times one is equal to one. Then we multiply the outer terms, and we get one times 𝑖, which is simply 𝑖. Next, we multiply the inner terms. Negative 𝑖 times one is negative 𝑖. And finally, we multiply the last term in each expression, and we get negative 𝑖 squared. Now, 𝑖 minus 𝑖 is zero, and we actually know that 𝑖 squared is equal to negative one. So this simplifies to one minus negative one. Well, that’s one plus one, which is just two.
But what do we do with our numerator? Well, we’re going to do this in stages. We’ll just begin by distributing the parentheses containing five plus five 𝑖 and eight minus four 𝑖. In the same way, we multiply the first terms to get 40. We multiply the outer terms. Five times negative four 𝑖 is negative 20𝑖. We have five 𝑖 times eight, which is 40𝑖, and then five 𝑖 times negative four 𝑖, which is negative 20𝑖 squared. Then negative 20𝑖 plus 40𝑖 is positive 20𝑖. But we can also rewrite negative 20𝑖 squared as negative 20 times negative one. Well, a negative times a negative is a positive. So we get 40 plus 20 as being 60. And then we find the product of five plus five 𝑖 and eight minus four 𝑖 is 60 plus 20𝑖.
Let’s now multiply all of this by one plus 𝑖. Distributing as before, and we get 60 plus 60𝑖 plus 20𝑖 plus 20𝑖 squared. 60𝑖 plus 20𝑖 is 80𝑖, but also 20 times 𝑖 squared is 20 times negative one. So it’s negative 20. And so we simplify this expression fully, and we get 40 plus 80𝑖. And so when we divide five plus five 𝑖 times eight minus four 𝑖 by one minus 𝑖, we get 40 plus 80𝑖 over two, which we can simplify further by dividing each term on the numerator by two. 40 divided by two is 20, and then 80𝑖 divided by two is 40𝑖. And so by multiplying the numerator and the denominator of our fraction by the conjugate of the denominator, we’ve expressed it in the form 𝑎 plus 𝑏𝑖. It’s 20 plus 40𝑖.