In Exercises 65–70, perform the indicated operation(s) and write ...

Question

In Exercises 65–70, perform the indicated operation(s) and write the result in standard form.

(4 - i)^2 - (1 + 2i)^2

Accepted Answer

Hello everyone in this video we're going to be looking at this practice problem where we need to simplify the expression five minus I squared minus two plus three I squared. And we need to make sure our answer is in standard form. So you can see that we're dealing with complex numbers because we have numbers that have imaginary components and also real components. So I'm gonna go ahead and write this out. So I have 5 -1 squared. I'm gonna expand that exponents. So five minus I squared is the same as five minus I times five minus I. And then I have minus two plus three. I squared. Can become two plus three I times two plus three. I so I'm gonna follow. Pen does recall that. Pen does gives us a sort of guideline into the steps that we should follow when simplifying expressions where ps parentheses. E. Is exponents and multiplication. D. Division, A addition and subtraction. So you can see that we have parentheses here. So I'm going to deal with those first and then deal with the subtraction later. So I'm going to start with my first set of parentheses. I have five times 5. 25. 5 times negative. I negative five I negative I times five negative five I. And negative I times negative. I is positive. I squared and notice I can combine the middle terms I'm left with 25 minus I plus I squared. And if I do the same to my second set of parentheses I have two times two is 42 times three I is six I three I times two is six I and three I times three I is nine I squared and I can do the same to my middle terms and combine them. So I'm left with four plus 12 I plus nine I squared. And there's a minus between these two. So now I'm gonna go ahead and distribute this minus or negative to all of these terms. So I'm left with 25 minus 10. I plus I squared minus four minus 12 I minus nine I squared. And you can see I can once again combine similar terms where I have 25 negative four becomes 21 then I have negative 10 I and negative 12 I. And that becomes negative 22 I and then I have a positive I squared and negative nine I squared. And that becomes negative eight I squared. So now it seems that I've simplified my expression the most but we have this weird I squared So recall that I is equal to the square root of -1. So if I do I squared, I have negative one times negative one. And whenever you have two square roots being multiplied and they have the same insights. Then we get back the inside components. So the square root of negative one times the square root of negative one gives me negative one. So you can see that by squaring I I get back a real number because negative one is a real number. So I'm gonna go ahead and go back to my expression and I'm gonna replace I squared with negative one. And if I do that I have 21 -22 I - times negative one. And you can see that this becomes 21 minus 22 I plus eight because negative eight times negative one is positive eight. And now I can combine my 21 and my eight and I'm off with 29 -22. And we can verify that this is in standard form because recall that standard form is the real component followed by the imaginary component and that is how we have it written. So this becomes my final answer and you can see that Answ. Choice D. is also 29 -22. I. So hopefully this video helped and see you next time.

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