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.

(2 + 3i)^3

Accepted Answer

Welcome back. I'm so glad you're here. We've got this expression three plus two. I all cubed and were to write it in standard form. So what is standard form. We've got this eye which means we're dealing with complex numbers. And the standard form for complex numbers is A plus B. I. And it's really close to standard form already except it's just got this three here. It's all being cubed. And wouldn't it be lovely if we could just distribute that three. But nope that's a common error. We have to re write this as three plus two I times three plus two I times three plus two. I. Yes this expression is cubed which means there's three of them and now we can just go ahead and start foiling. We can take our first three times three that's nine. Our Outsides three times two. I that's plus six. I our insides to I times three is again plus six. I. And our last two I times two. I that is four I squared. And we can combine our like terms. So those six eyes in the middle six I plus six. I Well that's going to give us 12 I. And we bring down that four I squared. But then we say hang on a second we recall from previous lessons that when we have I to the first that's just I but I to the second I squared that's negative one. When we have I to the third that's negative I. And when we have I to the fourth that is one. So this four times I squared here. Well that's four times negative one. So we've got nine plus 12. I plus four times negative one. All times three plus two. I we know that four times negative one. Well that is negative four. And now we can just combine like terms again. So we have 9 -4. Well that's five. So five plus 12 I times three plus two. I cool. We can foil this again. Our 1st 5 times three is 15. Our outsides five times 2. I is plus 10. I our insides 12 I times three is plus 36 I and our last 12 I times two I is plus 24. I square don't forget to square that. And now we've got a couple of things. We can combine our like terms or 10 I plus 36. I and we can simplify that. I squared so we've still got 15 now it's plus 46. I and 24 times I squared or 24 times negative one is going to be minus 24. We can again combine our like terms. So 15 minus 24 gives us a negative nine plus 46. I that's as simplified as it's going to get. And it is in our standard form that we're looking for. So we look at our answer choices and that matches with answer choice. D Well done. We'll catch you on the next one

In Exercises 61–64, write each complex number in standard form. (2 + 3i)^3

Key Concepts

Complex Numbers

Standard Form of Complex Numbers

Exponentiation of Complex Numbers

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