For Exercises 77-90, divide the complex numbers. Write the answer in the form a+b i. (see Exampl

step 1 Hi there. So today we're doing another complex number division. What we're going to need to know

For Exercises 77-90, divide the complex numbers. Write the...

Key Concepts

Division of Complex Numbers

This concept involves performing division where both the numerator and denominator are complex numbers. The typical method is to multiply the numerator and denominator by the complex conjugate of the denominator to eliminate the imaginary part, resulting in an expression that can be simplified to standard form.

Complex Conjugate

The complex conjugate of a complex number, given by a + bi, is defined as a - bi. It is used in division to cancel out the imaginary terms in the denominator when multiplied, making the denominator a real number.

Multiplication of Complex Numbers

Multiplying complex numbers requires using the distributive property and combining like terms, while applying the identity i^2 = -1. This operation is essential when multiplying by the complex conjugate and simplifying the result into its real and imaginary parts.

Standard Form of Complex Numbers

The standard form represents a complex number as a + bi, where 'a' is the real component and 'b' is the coefficient of the imaginary unit. This form is important because it clearly separates the real and imaginary parts, making further arithmetic or analysis more straightforward.

Transcript

00:01 Hi there.

00:01 So today we're doing another complex number division.

00:04 What we're going to need to know for this problem is that we have to simplify it to get it into a plus b .i form.

00:10 And we also have to use the foil method, which as we remember is just the first and then the outer, then the inner, then the last.

00:19 And i can represent that with the numerator multiplication in this problem.

00:23 So in order to begin this problem, we're going to go ahead and take the top and the bottom of the original equation times the opposite.

00:30 Of the bottom equation.

00:33 And in order to do that, we are just going to switch the signs.

00:36 So this still remains as negative 2, but then we do this.

00:39 We change the sign in the middle.

00:41 So it would be negative 2 plus 4i.

00:43 And then we do negative 2 plus 4i also times the numerator of the original equation.

00:48 And in order to get this, this is where the foil method comes in.

00:53 So in order for the foil method to work, we have to take the first, which is that 3 .4.

01:00 Times the negative 2, which is negative 6.

01:04 And then we get that three times the, so this is the outer values, which is the 3 times 4i, which is 12i.

01:12 And then we take the inner values times each other, which is 7i times negative 2, which is minus 14i.

01:19 And then we take the outer or the last two values, i should say, which are 7i and 4i, and that is just plus 28 i squared...

Please give Ace some feedback