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# Find all cubic roots of z = −1 + i

0
1147
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+1904

Find all cubic roots of $$z = −1 + i$$

Dec 22, 2015

#2
+109519
+10

Hi Gibsonj338

Find all cubic roots of

$$z=-1+i \mbox{ distance of z to the origin }=\sqrt{1^2+1^2}=\sqrt2\\ \mbox{cubic toots so there are 3 of them so they are } \frac{2\pi}{3} radians\; apart.\\ z=\sqrt2(cis\frac{3\pi}{4})\\ \sqrt[3]{z}=\sqrt[6]{2}(cis(\frac{3\pi}{4}\div3)),\quad\sqrt[6]{2}(cis((\frac{3\pi}{4}+2\pi)\div3)),\quad \sqrt[6]{2}(cis((\frac{3\pi}{4}+4\pi)\div3))\\ \sqrt[3]{z}=\sqrt[6]{2}(cis(\frac{3\pi}{4}\div3)),\quad\sqrt[6]{2}(cis(\frac{11\pi}{4}\div3)),\quad \sqrt[6]{2}(cis(\frac{19\pi}{4}\div3))\\ \sqrt[3]{z}=\sqrt[6]{2}(cis(\frac{3\pi}{12})),\quad\sqrt[6]{2}(cis(\frac{11\pi}{12})),\quad \sqrt[6]{2}(cis(\frac{19\pi}{12}))\\ \sqrt[3]{z}=\sqrt[6]{2}(cis(\frac{\pi}{4})),\quad\sqrt[6]{2}(cis(\frac{11\pi}{12})),\quad \sqrt[6]{2}(cis(\frac{19\pi}{12}))\\ or\$$

$$\sqrt[3]{z}=\frac{1+i}{\sqrt[3]{2}},\quad\sqrt[6]{2}(cos(\frac{11\pi}{12})+i\;sin(\frac{11\pi}{12})),\quad\sqrt[6]{2}(cos(\frac{19\pi}{12})+i\;sin(\frac{19\pi}{12}))\\ \sqrt[3]{z}=\frac{1+i}{\sqrt[3]{2}},\quad\sqrt[6]{2}(-cos(\frac{\pi}{12})+i\;sin(\frac{\pi}{12})),\quad\sqrt[6]{2}(cos(\frac{5\pi}{12})-i\;sin(\frac{5\pi}{12}))$$

Dec 23, 2015
edited by Melody  Dec 23, 2015

#1
+1904
+5

$$z =(−1 + i )^(1/3)$$

$$r=\sqrt((-1)^2+1^2)$$

$$r=\sqrt(1+1^2)$$

$$r=\sqrt(1+1)$$

$$r=\sqrt(2)$$

$$tan(\Theta)=1/-1$$

$$tan(\Theta)=-1$$

$$\Theta=tan^-1(-1)$$

$$\Theta=-\pi/4$$

$$\sqrt(2)e^(-\pi/4i)^(1/3)$$

$$\sqrt(2)^(1/3)e^(-\pi/12i)$$

$$\sqrt(2)^(1/3)*(cos(-\pi/12)+isin(-\pi/12))$$

$$\sqrt(2)^(1/3)*(0.965925826289 +isin(-\pi/12))$$

$$\sqrt(2)^(1/3)*(0.965925826289 +i*-0.258819045103)$$

$$\ 1.414213562373095)^(1/3)*(0.965925826289 +i*-0.258819045103)$$

$$\ 1.122462048309373*(0.965925826289 +i*-0.258819045103)$$

$$\ 1.084215081491274550528506797 +i*-0.290514555507789375384650419$$

$$\ 1.084215081491274550528506797 + -0.290514555507789375384650419i$$

Answer 1:$$\ 1.084215081491274550528506797 -0.290514555507789375384650419i$$

$$\sqrt(2)e^(7\pi/4i)^(1/3)$$

$$\sqrt(2)^(1/3)e^(7\pi/12i)$$

$$\sqrt(2)^(1/3)*(cos(7\pi/12)+isin(7\pi/12))$$

$$\sqrt(2)^(1/3)*(-0.258819045103 +isin(7\pi/12))$$

$$\sqrt(2)^(1/3)*(-0.258819045103 +i* 0.965925826289)$$

$$\ 1.414213562373095)^(1/3)*(-0.258819045103 +i* 0.965925826289)$$

$$\ 1.122462048309373*(-0.258819045103 +i* 0.965925826289)$$

$$\ -0.290514555507789375384650419 +i* 1.084215081491274550528506797$$

Answer 2:$$\ -0.290514555507789375384650419 +1.084215081491274550528506797i$$

$$\sqrt(2)e^(15\pi/4i)^(1/3)$$

$$\sqrt(2)^(1/3)e^(15\pi/12i)$$

$$\sqrt(2)^(1/3)*(cos(15\pi/12)+isin(15\pi/12))$$

$$\sqrt(2)^(1/3)*(-0.707106781187 +isin(15\pi/12))$$

$$\sqrt(2)^(1/3)*(-0.707106781187 +i* -0.707106781187)$$

$$\ 1.414213562373095)^(1/3)*(-0.707106781187 +i* -0.707106781187)$$

$$\ 1.122462048309373*(-0.793700525984607637192165751 +i* -0.793700525984607637192165751)$$

$$\ -0.793700525984607637192165751 +i*-0.793700525984607637192165751$$

$$\ -0.793700525984607637192165751 +-0.793700525984607637192165751i$$

Answer 3:$$\ -0.793700525984607637192165751 -0.793700525984607637192165751i$$

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Dec 22, 2015
edited by gibsonj338  Dec 22, 2015
#2
+109519
+10

Hi Gibsonj338

Find all cubic roots of

$$z=-1+i \mbox{ distance of z to the origin }=\sqrt{1^2+1^2}=\sqrt2\\ \mbox{cubic toots so there are 3 of them so they are } \frac{2\pi}{3} radians\; apart.\\ z=\sqrt2(cis\frac{3\pi}{4})\\ \sqrt[3]{z}=\sqrt[6]{2}(cis(\frac{3\pi}{4}\div3)),\quad\sqrt[6]{2}(cis((\frac{3\pi}{4}+2\pi)\div3)),\quad \sqrt[6]{2}(cis((\frac{3\pi}{4}+4\pi)\div3))\\ \sqrt[3]{z}=\sqrt[6]{2}(cis(\frac{3\pi}{4}\div3)),\quad\sqrt[6]{2}(cis(\frac{11\pi}{4}\div3)),\quad \sqrt[6]{2}(cis(\frac{19\pi}{4}\div3))\\ \sqrt[3]{z}=\sqrt[6]{2}(cis(\frac{3\pi}{12})),\quad\sqrt[6]{2}(cis(\frac{11\pi}{12})),\quad \sqrt[6]{2}(cis(\frac{19\pi}{12}))\\ \sqrt[3]{z}=\sqrt[6]{2}(cis(\frac{\pi}{4})),\quad\sqrt[6]{2}(cis(\frac{11\pi}{12})),\quad \sqrt[6]{2}(cis(\frac{19\pi}{12}))\\ or\$$

$$\sqrt[3]{z}=\frac{1+i}{\sqrt[3]{2}},\quad\sqrt[6]{2}(cos(\frac{11\pi}{12})+i\;sin(\frac{11\pi}{12})),\quad\sqrt[6]{2}(cos(\frac{19\pi}{12})+i\;sin(\frac{19\pi}{12}))\\ \sqrt[3]{z}=\frac{1+i}{\sqrt[3]{2}},\quad\sqrt[6]{2}(-cos(\frac{\pi}{12})+i\;sin(\frac{\pi}{12})),\quad\sqrt[6]{2}(cos(\frac{5\pi}{12})-i\;sin(\frac{5\pi}{12}))$$

Melody Dec 23, 2015
edited by Melody  Dec 23, 2015