In Exercises 21–28, divide and express the result in standard for...

Question

In Exercises 21–28, divide and express the result in standard form.

8i / 4−3i

Accepted Answer

Welcome back. Everyone in this problem. We want to perform the operation on the complex number 12, I divided by seven minus five I and put our answer in standard form for our answer choices. A is 30 37th plus 42 37th IB is negative 30 37th plus 42 37th. IC is 25 74th plus 74th I and D is negative 74th plus 35 74th. I. Now, what do we know here? Well, we are trying to divide 12 I by seven minus five I and we also want to put our answer in standard form. Well, recall that the standard form of a complex number is A plus B I where A is the real part and B is the imaginary part. So for us to convert this caution into standard form, we will need to multiply both the numerator and the denominator by the complex conjugate of the denominator. Let's break down all of what that means. No, you can also recall that the conjugate of a complex number conjugate of a complex number. Let's say A plus B I would be equal to the same real part A but the opposite imaginary part B. So the complex conjugate of seven minus five, I, the conjugate of seven minus five, I equals seven, the same real part plus five, I, the opposite imaginary part, we multiply the ent by seven plus five I because multiplying a value or multiplying the denominator by its complex conjugate turns it into a real number. And we also multiply the numerator by that value so that we can keep the value of the expression. So we are multiplying both of them by seven plus five. I. So let's go ahead and do that, try to turn our denominator into a real number and then see if we can express it in standard form. Now, starting with the numerator 12, I multiplied by seven equals 84 I and 12, I multiplied by five, I equals 60 I squared. In our denominator, there's something special about it. Notice we have seven minus five I and seven plus five I. So it's the sum and difference of the two same terms. And if you recall, we can use the difference of two squares to help us because recall that the difference of two squares says that if we are multiplying the sum and the difference of two expressions, it will be equal to the difference of their squares. In other words, A squared minus B squared, so I can rewrite seven minus five, I multiplied by seven plus five, I as seven squared minus five I squared, all squared. No, in my numerator, there's something else that can help us to simplify. Notice that 60 I squared can be written as a real number. Why? Well, again, recall that given I being equal to the square root of negative one, then I squared would be equal to the square root of negative one squared, which would just have been negative one. So I can apply that property to both my numerator and denominator. Because no, in my numerator, it will be 84 I plus 60 multiplied by negative one. And in my denominator seven squared is 49 5 I squared would be 25 I squared, which would have been 25 multiplied by negative one. Taking it a step further, we'll have 84 I minus 60 and 49 minus 25 multiplied by negative one. And that would be 49 plus 25 which would be 74. No, that means I can rewrite this as negative 60 divided by 74. The real number first plus 84 I divided by 74. The imaginary number after negative 60 74 have a common factor of two, two into negative 60 goes negative 30 times 2 to 74 goes 37 times. Likewise, 84 and 74 also have a common factor of 22 to goes 42 times and 20 to 74 goes 37 times. So I can write this as negative 30 37th plus 42 37th of I. No. Finally, my answer is in standard form and when we look back on our answer choices that would have been answer choice. B thanks a lot for watching everyone. I hope this video helped.

In Exercises 21–28, divide and express the result in standard form. 8i / 4−3i

Key Concepts

Complex Numbers

Division of Complex Numbers

Standard Form of Complex Numbers

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