In Exercises 1–10, perform the indicated operations and write the...

Question

In Exercises 1–10, perform the indicated operations and write the result in standard form.

3+4i / 4−2i

Accepted Answer

Welcome back. Everyone. In this problem, we want to simplify the expression three plus seven I all divided by six minus two I and express our answer in standard form. That is the typical form for a complex number for our answer choices. A is 1/10 minus six fifths of IB is 4/5 minus 9/10 of IC is 1/ plus six fifths of I and D is 4/5 plus 9/10 of I. Now, what do we already know? Well, we know that if we're dealing with a complex number, that is a caution, then recall, we can divide them where by multiplying the numerator and the denominator by the conjugate of the denominator. OK. And the conjugate of our denominator, oops conjugate, don't tell anybody. I didn't spell it right. The conjugate of six minus two, I is six plus two. I, it's just a number that has the same real part, but the opposite imaginary part. And it's important because if we multiply a number by its conjugate, the result will always be a real number. So to divide our two complex numbers, we can multiply the numerator. So that's three plus seven I and the denominator six minus two I by or conjugate, which is six plus two. I, so let's go ahead and do that three plus seven. I multiplied by six plus two. I, as a matter of fact, I don't have to write that part out. We can just go ahead and do the multiplication. I know you can three multiplied by six is 18 and three multiplied by two. I is six I seven, I multiplied by six is 42 I and seven. I multiplied by two, I is 14, I squared. No, in our denominator six multiplied by six is 36. 6, multiplied by two, I is 12. I negative two, I multiplied by six is negative 12 I and negative two, I multiplied by two, I is negative four I squared. Now also recall that is squared equals one equals negative one rather. So we can use that in our expression to help us solve because now we can rewrite it as 18 plus six. Well, six, I plus 42 I that's 48 I and no 14 I squared will overcome 14 multiplied by a negative one. Likewise in our denominator, we'll have 36 12 I minus 12 I those cancel and that will be minus four multiplied by negative one because I squared is negative one, no, 18, 14 multiplied by negative one is negative 14 and 18 minus 14 is four. So our N numerator will have four plus 48 I and in our denominator, we'll have 36 minus four times negative one. So that becomes 36 plus four, which would be 40. Now, we can divide both terms in numerator by the denominator and four divided by 40 is one. So over here, it would be 1/10 and 48 I divided by 40. Well, that would work out to be six fifths of I. Therefore, our answer is 1/10 plus six fifths of I, which that would be answer choice. C Thanks for watching everyone. I hope this video helped.

In Exercises 1–10, perform the indicated operations and write the result in standard form. 3+4i / 4−2i

Key Concepts

Complex Numbers

Division of Complex Numbers

Standard Form of Complex Numbers

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