+0  
 
0
558
4
avatar

Please help with this: (5 - 3i)^1/3. Calculate as many forms as possible. I thank you.

advanced
Guest Dec 21, 2015

Best Answer 

 #4
avatar+93691 
+5

 (5 - 3i)^1/3.

 

\(r = mod z = |z| = \sqrt{5^2+(-3)^2}=\sqrt{34}\)

 

\(5-3i=\sqrt{34}(\frac{5}{\sqrt{34}}-\frac{3i}{\sqrt{34}})\\ 4th\;quad\\ argument \;of\; z = arg(z) =\theta=2\pi-acos(\frac{5}{\sqrt{34}})\\ \theta\approx 5.7427658\;radians\\ 5-3i=\sqrt{34}\;e^{5.7427658i} \)
 

 

\(|z^{1/3}| = \sqrt{34}^{1/3}=\sqrt[6]{34}\)

 

\(1)  arg(z^{1/3})=5.7427658/3 = 1.914255\\ 2)  arg(z^{1/3}) = 1.914255 +\frac{2 \pi}{3} = 4.008650\\ 3)  arg(z^{1/3}) = 1.914255 +2*\frac{2 \pi}{3}= 6.103045\\\)

 

So the 3 cubed roots of are 5-3i  are

\(\sqrt[6]{34}*e^{1.914255i}\quad and \quad \sqrt[6]{34}*e^{4.008650i}\quad and \quad \sqrt[6]{34}*e^{6.103045i}\)

 

cos(1.914255) = -0.336745771286

sin(1.914255) = 0.94159560615

\(1st\;root=\sqrt[6]{34}\;(-0.336746\;+\;0.941596\;i)\)

etc

check the first root

(34^(1/6)*e^(1.914255*i))^3 = 4.9999975792700509-3.0000040345460377i      Near enough    laugh

 

 

 

Melody  Dec 25, 2015
 #2
avatar
+5

z = (5 - 3i)^(1/3)

Algebraic form:
z = 1.7707675-0.3224815i

Exponential form:
z = 1.7998922 × ei (-10°19'16″)

Trigonometric form:
z = 1.7998922 × (cos (-10°19'16″) + i sin (-10°19'16″))

Polar form:
r = |z| = 1.79989
φ = arg z = -10.32125° = -10°19'16″ = -0.05734π
 

THIS IS THE ACCURATE SOLUTION. THE FIRST ANSWER CALCULATED (5-3i)^1*3

Guest Dec 22, 2015
 #3
avatar+93691 
0

Could you talk us through it please ?

Melody  Dec 22, 2015
 #4
avatar+93691 
+5
Best Answer

 (5 - 3i)^1/3.

 

\(r = mod z = |z| = \sqrt{5^2+(-3)^2}=\sqrt{34}\)

 

\(5-3i=\sqrt{34}(\frac{5}{\sqrt{34}}-\frac{3i}{\sqrt{34}})\\ 4th\;quad\\ argument \;of\; z = arg(z) =\theta=2\pi-acos(\frac{5}{\sqrt{34}})\\ \theta\approx 5.7427658\;radians\\ 5-3i=\sqrt{34}\;e^{5.7427658i} \)
 

 

\(|z^{1/3}| = \sqrt{34}^{1/3}=\sqrt[6]{34}\)

 

\(1)  arg(z^{1/3})=5.7427658/3 = 1.914255\\ 2)  arg(z^{1/3}) = 1.914255 +\frac{2 \pi}{3} = 4.008650\\ 3)  arg(z^{1/3}) = 1.914255 +2*\frac{2 \pi}{3}= 6.103045\\\)

 

So the 3 cubed roots of are 5-3i  are

\(\sqrt[6]{34}*e^{1.914255i}\quad and \quad \sqrt[6]{34}*e^{4.008650i}\quad and \quad \sqrt[6]{34}*e^{6.103045i}\)

 

cos(1.914255) = -0.336745771286

sin(1.914255) = 0.94159560615

\(1st\;root=\sqrt[6]{34}\;(-0.336746\;+\;0.941596\;i)\)

etc

check the first root

(34^(1/6)*e^(1.914255*i))^3 = 4.9999975792700509-3.0000040345460377i      Near enough    laugh

 

 

 

Melody  Dec 25, 2015

12 Online Users

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.