Given z=3CiS(4π5)z=3CiS\left(\frac{4\pi}{5}\right), calculate z5z^5.

z 5 = 243 C i S ( 4 π ) z^5=243CiS\left(4\pi\right)

Given z = 3 C i S ( 4 π 5 ) z=3CiS\left(\frac{4\pi}{5}\right) , calculate z 5 z^5 .

this problem, we are given a complex number in polar form, and we are asked to evaluate what this complex number would be if we raised it to the 5th power. Now to find z to the 5th power, recall that we can use de Moivre's theorem that says we need to take the r value and raise it to whatever power this is, and then multiply this number by whatever the angle is. So we're going to have 3, which is this number here, and we're going to raise that to a power of 5, because 5 is the exponent we're dealing with, and that's going to be multiplied by Sys, and again this is this abbreviated version for the Sines and Cosines, then we won't have our exponent 5 being multiplied by this angle 4pi over 5 radians. So if we do this, all we really need to do is simplify to find our solution. Now 3 to the 5th power, this is something that you can just plug into your calculator. You could evaluate 3 times 3 times 3 times 3, which would be 3 to the 5th power, but this can be a bit get a bit tedious. So if you plug this into your calculator, you should get 243. So this term right here is going to be 243, and then that's going to be cis, and then we have 5 times 4pi over 5 radians. Now this 5 will cancel with that 5 leaving us with just 4pi radians. So that right there is going to be the solution for z to the 5th, and that is how you can find this complex power with this complex number raised to the 5th power. So I hope you found this video helpful thanks for watching.

Given z = 3 C i S ( 4 π 5 ) z=3CiS\left(\frac{4\pi}{5}\right) , calculate z 5 z^5 .

z 5 = 243 C i S ( 4 π ) z^5=243CiS\left(4\pi\right)

z 5 = 243 C i S ( 32 π ) z^5=243CiS\left(32\pi\right)

z 5 = 243 C i S ( 100 ) z^5=243CiS\left(100\right)

z 5 = 243 C i S ( 4 ) z^5=243CiS\left(4\right)

11. Graphing Complex Numbers

Powers of Complex Numbers (DeMoivre's Theorem)

Trigonometry

11. Graphing Complex Numbers

Powers of Complex Numbers (DeMoivre's Theorem)

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Multiple Choice

Given $z = 3 C i S (\frac{4 π}{5})$ , calculate $z^{5}$ .

$z^{5} = 243 C i S (4 π)$

$z^{5} = 243 C i S (32 π)$

$z^{5} = 243 C i S (100)$

$z^{5} = 243 C i S (4)$

Verified step by step guidance

Identify the given complex number in polar form: z = 3 \text{CiS}\left(\frac{4\pi}{5}\right). Here, the modulus is 3 and the argument is \frac{4\pi}{5}.

To find z^5, use De Moivre's Theorem, which states that for a complex number z = r \text{CiS}(\theta), z^n = r^n \text{CiS}(n\theta).

Calculate the modulus of z^5: (3)^5 = 243.

Calculate the argument of z^5: 5 \times \frac{4\pi}{5} = 4\pi.

Express the result in polar form: z^5 = 243 \text{CiS}(4\pi).

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