Write each expression in the standard form a+b i (1-i)^5 |...

Name: Problem 51, Chapter 10: Algebra and Trigonometry
Uploaded: 2020-06-18T12:59:52-08:00
Duration: 3 min 6 s
Channel: Joseph Lentino
Description: Write each expression in the standard form a+b i (1-i)^5

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Key Concepts

Standard Form of Complex Numbers

This concept deals with expressing any complex number in the form a + bi, where a represents the real part and b represents the imaginary part. This form is fundamental in complex number arithmetic, as it allows for straightforward addition, subtraction, and comparison of complex numbers, as well as facilitating visualization on the complex plane.

Binomial Theorem

The Binomial Theorem is a powerful algebraic tool used to expand expressions raised to a power, such as (x + y)^n. When applied to complex numbers, it provides a systematic method to expand expressions like (a + bi)^n by separating the terms into real and imaginary parts, which can then be combined to yield the standard form of the complex number.

De Moivre's Theorem

De Moivre's Theorem connects complex numbers and trigonometry by allowing the computation of powers and roots of complex numbers represented in polar form. By converting a complex number from its standard form to polar form, raising it to a power, and then converting it back, one can efficiently find the expression of the result in a + bi form, which is particularly useful for complex exponentiation.

Transcript

00:02 Okay, so we need our answer in a plus b -i form, and we've got one minus i to the fifth.

00:08 The easiest way to work with this, instead of multiplying the whole thing out and trying to find the like terms, et cetera, is to switch this into polar form.

00:16 So the first thing that i always do is kind of figure out where this is going to be, which is right here.

00:23 We have a cosine of one.

00:26 We have a sign of negative one.

00:30 So this is going to be a 45 degree angle here.

00:33 It's in the fourth quadrant.

00:35 So this problem is really the same as, oh, and by the way, one squared plus one squared, take the square root.

00:41 It's going to be root 2.

00:43 This is the same as me saying root 2, cosine i sign of 315 degrees.

00:51 And for those people that aren't familiar with cosine, i sign, that's the same as me saying cosine of 315 plus i sine, just to simplify a little bit of space.

01:03 Because obviously it's going to be the same answer for both.

01:08 And this is all to the fifth power.

01:10 So i'm going to take root 2 to the fifth power, which is going to come out to be 4 root 2.

01:17 And then i have to take cosine and i sign and take 315 and multiply that times 5.

01:26 When i multiply that times 5, i get cosine i sign of 1 ,5755...

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