Video Transcript
Simplify eight plus seven 𝑖 multiplied by one plus eight 𝑖 over six plus four 𝑖.
In order to simplify this fraction, we’re going to perform two individual steps. We’ll begin by finding the product of the expressions on the numerator of this fraction. That’s eight plus seven 𝑖 multiplied by one plus eight 𝑖. Next, we’ll take this expression and we’ll divide it by the second complex number, six plus four 𝑖.
So let’s begin by distributing the parentheses on the numerator of our fraction. There are of course a number of ways that we can do this. We’re going to use the FOIL method. That is, we’re going to multiply the first term in the first expression by the first term in the second. Eight multiplied by one is eight. Then, we’re going to multiply the outer two terms. That’s eight multiplied by eight 𝑖, which is 64𝑖. Next, we multiply the inner terms, seven 𝑖 multiplied by one, and that’s seven 𝑖. Finally, we multiply the last term in each expression. Here, that’s seven 𝑖 multiplied by eight 𝑖, which is 56𝑖 squared.
Now, of course we can collect like terms, and we see that 64𝑖 plus seven 𝑖 is 71𝑖. But we’re also able to evaluate this final term. That’s 56𝑖 squared. We know that 𝑖 itself is the solution to the equation 𝑥 squared equals negative one. In other words, 𝑖 squared is equal to negative one. And so 56𝑖 squared is 56 multiplied by negative one, which is negative 56. And so our expression becomes eight plus 71𝑖 minus 56, which is simply negative 48 plus 71𝑖.
So we’ve completed the first step. We’ve found the product of the expressions on the numerator of our fraction. We said that the next thing we’re going to do is take that expression and divide it by the expression on the denominator of our fraction. So we’re going to calculate negative 48 plus 71𝑖 divided by six plus four 𝑖. So, how do we divide by a complex number?
Well, to divide by a complex number, we multiply both the numerator and denominator of the expression by the conjugate of the denominator. Let’s remind ourselves what that means. Say we have a general complex number 𝑧 equals 𝑎 plus 𝑏𝑖, where of course 𝑎 and 𝑏 must be real numbers. The complex conjugate of 𝑧, which we usually denote as 𝑧 bar or sometimes 𝑧 star, is 𝑎 minus 𝑏𝑖. So we find the complex conjugate by simply changing the sign of the imaginary part. This means that the complex conjugate of the denominator of our fraction and, thus, the expression we’re going to multiply through by is six minus four 𝑖.
And then we can perform a very similar process to before. We’re going to multiply out the numerators and then separately the denominators. Negative 48 multiplied by six is negative 288. Then, multiplying negative 48 by negative four 𝑖, we get positive 192𝑖. 71 multiplied by six is 426. So we have 426𝑖. And 71𝑖 multiplied by negative four 𝑖 is negative 284𝑖 squared.
Let’s repeat this process with the denominator. When we do, we get 36 minus 24𝑖 plus 24𝑖 minus 16𝑖 squared. And the reason that we multiplied by that complex conjugate is because we can now cancel out the 𝑖 terms. And of course we know that negative 16𝑖 squared will be equal to a real number since 𝑖 squared is negative one. Replacing 𝑖 squared with negative one then, the numerator becomes negative 288 plus 618𝑖 plus 284. And then the denominator just becomes 36 plus 16.
This simplifies to negative four plus 618𝑖 all over 52, which can then be split into two separate fractions as shown. Once we have this, we can then divide both the numerator and the denominator in each part by the highest common factor of each. And we find that when we simplify our fraction, we get negative one-thirteenth plus 309 over 26 𝑖.
