In Exercises 95–99, perform the indicated operations and write th...

Question

In Exercises 95–99, perform the indicated operations and write the result in standard form.

8/(1 + 2/i)

Accepted Answer

Welcome back. I'm so glad you're here. We've got this expression we have seven divided by and then in the denominator we've got one plus and then another fraction three divided by I. We are to simplify and write the answer in standard form. Well what is Standard form? We've got complex numbers here we're dealing with an eye. And Standard form is A. Plus B. I. And one of the first steps in getting it into standard form is that we want to clear any I. Values out of denominators. And we've actually got an I. Value inside a denominator, inside a denominator. So we first got to work with this three divided by I. That fraction and clear the out of that denominator first. So how do we do that? We recall from previous lessons that in order to clear and I. Value out of a denominator we multiply both the numerator and the denominator by the complex conjugate. What is the complex conjugate? The complex conjugate is when you take an I. Value and it could be positive or negative but you're going to copy over the A. And the B. I. And then you're just going to switch the sign. So if you started off with a positive in between your A. Plus B. I. You'll now have a negative and that's great. Usually you're going to see things that look just like A plus B. I. But this time we just have an eye and that doesn't really look like standard form. But let me show you how that works. We do have an eye value and it's like there's an invisible one in front of it and it is positive in this case and are A here is just a zero. So that's what we've got right now. This i is the same as zero plus one eye. And we're going to multiply that times okay. Keeping the A value, keeping the zero, keeping the B. Value, keeping the one and keeping the eye and then just changing the sign. And how does that simplify 0 -1. I well that is just negative. I so we are going to take this little fraction here and we're going to multiply it by negative i in the numerator and the denominator. And when we bring this over and then multiply straight across three times negative I is now negative three I I times negative I is going to be a negative I squared. Now we can do a couple of things here. We can recall from previous lessons that by definition I squared equals negative one. So simplifying this a bit. We've got seven divided by one plus negative three. I over negative negative one. So seven divided by one plus negative three. I over one times negative times negative one is just one and then we can drop that divided by one and we've just got 7/1 plus negative three. I well that's just minus three. I great. We have got the I value cleared from one of our denominators but we still got an I value in our other denominator here. So now we've got to go through this process again. We've got a negative in here this time so we can think of it as a minus B. I. So one minus three I. And getting our complex conjugate. We keep the A. Value, keep the one, keep the B. Value, that's the three and the eye and flip the sign. So it's now one plus three. I we're going to multiply both the numerator and the denominator by one plus three. I. And multiplying in that numerator, distributing the 77 times one is 77 times three. I is plus 21. I. And now we could go ahead and foil out the denominator but we can recall from previous lessons that when you multiply a number by its complex conjugate, you're going to get a squared plus B squared. and so identifying are a value here. A is one And B is three. So we've got one squared plus three squared or in other words one plus nine that equals 10. So our new denominator is just 10 and this is almost in standard form we got the I value out of the denominator. Yay, now we just have to separate this so that our I value is distinct from our non I value. So 7/10 plus 21 10th I we look at our answer choices and that matches with answer choice A Well done. We'll catch you on the next one.

In Exercises 95–99, perform the indicated operations and write the result in standard form. 8/(1 + 2/i)

Key Concepts

Complex Numbers

Standard Form of Complex Numbers

Rationalizing the Denominator

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