Find all the complex roots. Write roots in rectangular form. If necessary, round to the nearest

Find all the complex roots. Write roots in rectangular form....

Key Concepts

nth Roots of Complex Numbers

The process for finding all nth roots of a complex number involves expressing the number in polar form and applying De Moivre's Theorem. The roots are given by taking the nth root of the magnitude and dividing the argument plus any integer multiple of 2? by n, leading to n distinct solutions spaced evenly around the circle.

De Moivre's Theorem

De Moivre's Theorem states that for any complex number in polar form and any integer n, (r(cos ? + i sin ?))^n equals r^n times (cos(n?) + i sin(n?)). This theorem is essential for finding the nth roots of complex numbers by evenly distributing the angle around the circle.

Conversion to Rectangular Form

Once a complex number's roots are determined in polar form, conversion to rectangular form involves computing the cosine and sine of the angle components to obtain the real and imaginary parts. This conversion is useful for presenting the final answers in the standard a + bi format.

Complex Numbers

Complex numbers extend the real numbers by including an imaginary unit i, where i² = -1. They are used to express solutions to equations that have no real solutions, representing numbers as a combination of a real part and an imaginary part.

Polar Form

The polar form of a complex number expresses it in terms of a magnitude (or absolute value) and an angle (or argument) with respect to the positive real axis. This form is particularly useful in simplifying multiplication, division, and the extraction of roots of complex numbers.

Transcript

00:04 We are asked to find the fifth roots of 32, and they don't just want two.

00:11 They want all of the complex roots as well.

00:15 That's why we have the formula there at the top of the screen.

00:19 That formula finds us all of the complex roots of any number, even a number that just appears to be a real number.

00:28 First thing we need to do is turn the 32 into polar form.

00:34 So obviously our radius is 32.

00:36 Since it's just a real number, if we were to graph this, we would be plotting this out on the real axis at 32.

00:45 And obviously our distance from the origin out to there is 32.

00:49 And the angle that we're making with the positive axis is zero.

00:53 So this is going to be 32 times the cosine of zero degrees plus i times a sign of 0 degrees, plus i times a sign of 0.

01:04 Degrees and i'm actually going to change my formula up here.

01:07 I have this formula written out from a previous problem where we were using radiance.

01:14 So instead of two pi, this should be 360 if we're doing it in degrees.

01:22 That's the only difference between the two formulas is, is it two pi or is it 360 that you're working with? i'm going to work with this in radiance so i'm going to change that to 360 degrees.

01:34 Now, when we're doing the fifth root, that five is the index here on our radical.

01:43 So that means that n equals five, which means we're going to use a k value in this formula from zero up to five minus one, so from zero to four.

01:54 So that means we're going to have five answers, starting with a value of k equals zero, k equals one, k equals two, k equals three, and then k equals four.

02:07 That's your rule of thumb you want to remember on all of these problems that you are always going to be doing the same number of roots as the index that you are given to work with.

02:22 So, k equals zero.

02:24 Well, actually, all of these start off with the nth root of your argument, the nth root of r.

02:30 And as i mentioned already, the fifth root of 32 is two.

02:34 So all these are going to start with a two.

02:36 And the only difference is going to be the angle measurement that we have here.

02:42 So the angle measurement is theta plus 360 times k divided by n.

02:50 If k is zero, then there's nothing to add to theta.

02:53 So we're just taking our angle, dividing it by our value of n, which is five, and zero divided by five is zero.

03:01 So for our first answer, we're going to have two times a cosine of zero degrees plus, i times the sign of zero degrees.

03:13 Now for the rest of these, we are adding 360 divided by n for every k value that we increase...

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