In Exercises 1–3, perform the indicated operations and write the ...

Question

In Exercises 1–3, perform the indicated operations and write the result in standard form.

5 / 2−i

Accepted Answer

Welcome back. Everyone. In this problem, we want to simplify the expression 40 divided by three minus I and express the answer in standard form if necessary for our answer choices, A is six negative six fifths plus 2/5 of IB is six fifths minus 2/5 of I C is 12 plus four, I and D is 12 minus four. I. Now, what do we already know? Well, we know that we are dividing by a complex number and to divide by a complex number, we can use the conjugate of that complex number. Recall that the conjugate of a complex number. So let's say the conjugate of A plus B I would be equal to A minus B I. In other words, we keep the real part and use the opposite of the imaginary part of the number. So that means the conjugate of our complex number in the denominator, which is three minus I would be three plus I. So if we want to divide by that complex number, we can multiply the numerator and denominator of the co by the conjugate. So let's do that. That means 40 divided by three minus I, you're going to be multiplied by three plus I and by by three plus I. No, when we go ahead and multiply 40 multiplied by three is 40 multiplied by I is 40. I. In our denominator notice that both of these are the difference of two squares, right? Three minus I multiplied by three plus I. So the result would be three squared minus I squared. No, we can turn our denominate into a real number if we recall the property that I squared equals negative one. So in that case, our problem now becomes 120 plus 40. I divided by three squared, which is nine minus I squared, which is negative one, nine minus negative one becomes nine plus one, which is 10. And if we divide both our terms by 10, 120 divided by 10 is 12 and 40 I divided by 10 is four. I therefore our answer is 12 plus four. I which would have been answer choice. C Thanks a lot for watching. I hope this video helped.

In Exercises 1–3, perform the indicated operations and write the result in standard form. 5 / 2−i

Key Concepts

Complex Numbers

Standard Form of Complex Numbers

Rationalizing the Denominator

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