+0

# Complex operation

0
327
4

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
+90996
+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

Melody  Dec 25, 2015
Sort:

### 3+0 Answers

#2
+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
+90996
0

Could you talk us through it please ?

Melody  Dec 22, 2015
#4
+90996
+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

Melody  Dec 25, 2015

### 4 Online Users

We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details