In Exercises 61–64, write each complex number in standard form.

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Question

In Exercises 61–64, write each complex number in standard form.

(1 + i)^3

Accepted Answer

Hello everyone in this video we're going to be looking at this practice problem where we want to solve or do the expression two plus I two the third and we want to make sure our answer is in standard form so you can see here we have a complex number two plus I we have the real component to in the imagine Neri component I and it's to the third power. So if I write that out I have two plus I times two plus I times two plus I. So in this case when we multiply complex numbers it's very similar to multiplying polynomial. Where you just want to make sure that every factor in one set of parentheses is multiplied by the factors in the other set of parentheses. So let's take this two parentheses out of time. So dealing with these first two parentheses here I have two times two is 42 times I is too I I times two is 2 I and I times I is I squared. And notice how these middle terms can be combined. So I'm left with four plus four I plus I squared. And in this case recall that I is equal to the square root of -1. So if I have I squared I have the square root of negative one times the square root of negative one. Which leaves me with the square root of negative one squared. So I can pull out the negative one. Yeah so this tells me that I squared is equal to negative one. So I'm going to go back to my expression and whenever I see and I squared I can just write negative one. So this becomes four plus four I plus negative one. And if I simplify this all the way I have three plus four. I but notice I still need to deal with this third two plus I. So we're gonna do the multiplication same as before. Three times two is 63 times I is three I four I times two is eight. I. And four I times I is for I squared notice we have the I squared again so I'm gonna go ahead and substitute the negative one again combining these middle terms. I have six plus 11. I Plus four times negative one. So I have two plus 11. I I got two because four times negative one is negative four and six minus four is two. And we want to make sure the answer is in standard form and in this case it is because we have the real number two. We're in first followed by the imaginary number 11. I So this becomes my final answer for this expression. And you can see that this aligns with answer choice B. So hopefully this video helped and see you next time

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