### Video Transcript

Simplify eight plus seven π multiplied by one plus eight π over six plus four π.

In order to simplify this fraction, weβre going to perform two individual steps. Weβll begin by finding the product of the expressions on the numerator of this fraction. Thatβs eight plus seven π multiplied by one plus eight π. Next, weβll take this expression and weβll divide it by the second complex number, six plus four π.

So letβs begin by distributing the parentheses on the numerator of our fraction. There are of course a number of ways that we can do this. Weβre going to use the FOIL method. That is, weβre going to multiply the first term in the first expression by the first term in the second. Eight multiplied by one is eight. Then, weβre going to multiply the outer two terms. Thatβs eight multiplied by eight π, which is 64π. Next, we multiply the inner terms, seven π multiplied by one, and thatβs seven π. Finally, we multiply the last term in each expression. Here, thatβs seven π multiplied by eight π, which is 56π squared.

Now, of course we can collect like terms, and we see that 64π plus seven π is 71π. But weβre also able to evaluate this final term. Thatβs 56π squared. We know that π itself is the solution to the equation π₯ squared equals negative one. In other words, π squared is equal to negative one. And so 56π squared is 56 multiplied by negative one, which is negative 56. And so our expression becomes eight plus 71π minus 56, which is simply negative 48 plus 71π.

So weβve completed the first step. Weβve found the product of the expressions on the numerator of our fraction. We said that the next thing weβre going to do is take that expression and divide it by the expression on the denominator of our fraction. So weβre going to calculate negative 48 plus 71π divided by six plus four π. 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. Letβs remind ourselves what that means. Say we have a general complex number π§ equals π plus ππ, where of course π and π must be real numbers. The complex conjugate of π§, which we usually denote as π§ bar or sometimes π§ star, is π minus ππ. 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 weβre going to multiply through by is six minus four π.

And then we can perform a very similar process to before. Weβre 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 π, we get positive 192π. 71 multiplied by six is 426. So we have 426π. And 71π multiplied by negative four π is negative 284π squared.

Letβs repeat this process with the denominator. When we do, we get 36 minus 24π plus 24π minus 16π squared. And the reason that we multiplied by that complex conjugate is because we can now cancel out the π terms. And of course we know that negative 16π squared will be equal to a real number since π squared is negative one. Replacing π squared with negative one then, the numerator becomes negative 288 plus 618π plus 284. And then the denominator just becomes 36 plus 16.

This simplifies to negative four plus 618π 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 π.