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Find all of the fifth roots of the complex number \(4+32i\).  Put your answers in the rectangular form \(a+bi\) and in the polar form \(z=re^(i\Theta )\).  Please show how you got to your answers.

 Nov 24, 2015

Best Answer 

 #2
avatar+1884 
+10

\((4+32i)^(1/5)\)

 

\(r=\sqrt(a^2+b^2)\)

 

\(r=\sqrt(4^2+32^2)\)

 

\(r=\sqrt(16+1024)\)

 

\(r=\sqrt1040\)

 

\(r=4\sqrt65\)

 

\(tan(\Theta)=b/a\)

 

\(tan(\Theta)=32/4\)

 

\(tan(\Theta)=8\)

 

\(\Theta=tan^-1(8)\)

 

\(\Theta ≈1.4464413322481\)

 

 

\(z=r*e^(i*\Theta)\)

 

\(z≈4\sqrt65*e^(i*1.4464413322481)\)

 

\(z^(1/5)≈(4\sqrt65*e^(i*1.4464413322481))^(1/5)\)

 

\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*1.4464413322481*(1/5))\)

 

\(z^(1/5)≈2.003103242348*e^(i*1.0766143512748)\)

 

\(z=r*(cos(\Theta)+i*sin(\Theta))\)

 

\(z≈2.003103242348*(cos(1.0766143512748)+i*sin(1.0766143512748))\)

 

\(z≈2.003103242348*(0.4743116564868+i*0.88035700289064)\)

 

\(z≈0.9500952169545+i*1.763445966914\)

 

\(z≈0.9500952169545+1.763445966914i\)

 

 

\(z^(1/5)≈(4\sqrt65*e^(i*7.7296266394277))^(1/5)\)

 

\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*7.7296266394277*(1/5))\)

 

\(z^(1/5)≈2.003103242348*e^(i*1.5459253278855)\)

 

\(z=r*(cos(\Theta)+i*sin(\Theta))\)

 

\(z≈2.003103242348*(cos(1.5459253278855)+i*sin(1.5459253278855))\)

 

\(z≈2.003103242348*(0.024868434927217+i*0.99969073264899)\)

 

\(z≈-0.049814942634829+i*0.0001545372385216\)

 

\(z≈-0.049814942634829+0.0001545372385216i\)

 

 

\(z^(1/5)≈(4\sqrt65*e^(i*14.012811946607))^(1/5)\)

 

\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*14.012811946607*(1/5))\)

 

\(z^(1/5)≈2.003103242348*e^(i*1.6955283616309)\)

 

\(z=r*(cos(\Theta)+i*sin(\Theta))\)

 

\(z≈2.003103242348*(cos(1.6955283616309)+i*sin(1.6955283616309))\)

 

\(z≈2.003103242348*(-0.12440885450163+i*0.99223104009177)\)

 

\(z≈-0.24920377983349+i*1.9875412136019\)

 

\(z≈-0.24920377983349+1.9875412136019i\)

 

 

\(z^(1/5)≈(4\sqrt65*e^(i*20.2959972538))^(1/5)\)

 

\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*20.2959972538*(1/5))\)

 

\(z^(1/5)≈2.003103242348*e^(i*4.059994507574)\)

 

\(z=r*(cos(\Theta)+i*sin(\Theta))\)

 

\(z≈2.003103242348*(cos(4.05994507574)+i*sin(4.05994507574))\)

 

\(z≈2.003103242348*(-0.60772245598411+i*-0.7941494925344)\)

 

\(z≈-1.2173308220295+i*1.5907634234047\)

 

\(z≈-1.2173308220295+1.5907634234047i\)

 

 

\(z^(1/5)≈(4\sqrt65*e^(i*26.5918256098))^(1/5)\)

 

\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*26.5918256098*(1/5))\)

 

\(z^(1/5)≈2.003103242348*e^(i*5.315836512196)\)

 

\(z=r*(cos(\Theta)+i*sin(\Theta))\)

 

\(z≈2.003103242348*(cos(5.315836512196)+i*sin(5.315836512196))\)

 

\(z≈2.003103242348*(0.56748448302716+i*-0.82338409112843)\)

 

\(z≈-1.1367300079339+i*-1.6493233426371\)

 

\(z≈-1.1367300079339+-1.6493233426371i\)

 

\(z≈-1.1367300079339-1.6493233426371i\)

.
 Nov 26, 2015
 #1
avatar
+5

z = (4 + 32i)^(1/5)  

Divide: 1 / 5 =0.2

Power: (4+32i) ^ 0.2 = 1.9198686+0.5714255i


Algebraic form:
z = 1.9198686+0.5714255i

Exponential form:
z = 2.0031032 × ei 16°34'30″

Trigonometric form:
z = 2.0031032 × (cos 16°34'30″ + i sin 16°34'30″)

Polar form:
r = |z| = 2.0031
φ = arg z = 16.575° = 16°34'30″ = 0.09208π

 Nov 24, 2015
 #2
avatar+1884 
+10
Best Answer

\((4+32i)^(1/5)\)

 

\(r=\sqrt(a^2+b^2)\)

 

\(r=\sqrt(4^2+32^2)\)

 

\(r=\sqrt(16+1024)\)

 

\(r=\sqrt1040\)

 

\(r=4\sqrt65\)

 

\(tan(\Theta)=b/a\)

 

\(tan(\Theta)=32/4\)

 

\(tan(\Theta)=8\)

 

\(\Theta=tan^-1(8)\)

 

\(\Theta ≈1.4464413322481\)

 

 

\(z=r*e^(i*\Theta)\)

 

\(z≈4\sqrt65*e^(i*1.4464413322481)\)

 

\(z^(1/5)≈(4\sqrt65*e^(i*1.4464413322481))^(1/5)\)

 

\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*1.4464413322481*(1/5))\)

 

\(z^(1/5)≈2.003103242348*e^(i*1.0766143512748)\)

 

\(z=r*(cos(\Theta)+i*sin(\Theta))\)

 

\(z≈2.003103242348*(cos(1.0766143512748)+i*sin(1.0766143512748))\)

 

\(z≈2.003103242348*(0.4743116564868+i*0.88035700289064)\)

 

\(z≈0.9500952169545+i*1.763445966914\)

 

\(z≈0.9500952169545+1.763445966914i\)

 

 

\(z^(1/5)≈(4\sqrt65*e^(i*7.7296266394277))^(1/5)\)

 

\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*7.7296266394277*(1/5))\)

 

\(z^(1/5)≈2.003103242348*e^(i*1.5459253278855)\)

 

\(z=r*(cos(\Theta)+i*sin(\Theta))\)

 

\(z≈2.003103242348*(cos(1.5459253278855)+i*sin(1.5459253278855))\)

 

\(z≈2.003103242348*(0.024868434927217+i*0.99969073264899)\)

 

\(z≈-0.049814942634829+i*0.0001545372385216\)

 

\(z≈-0.049814942634829+0.0001545372385216i\)

 

 

\(z^(1/5)≈(4\sqrt65*e^(i*14.012811946607))^(1/5)\)

 

\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*14.012811946607*(1/5))\)

 

\(z^(1/5)≈2.003103242348*e^(i*1.6955283616309)\)

 

\(z=r*(cos(\Theta)+i*sin(\Theta))\)

 

\(z≈2.003103242348*(cos(1.6955283616309)+i*sin(1.6955283616309))\)

 

\(z≈2.003103242348*(-0.12440885450163+i*0.99223104009177)\)

 

\(z≈-0.24920377983349+i*1.9875412136019\)

 

\(z≈-0.24920377983349+1.9875412136019i\)

 

 

\(z^(1/5)≈(4\sqrt65*e^(i*20.2959972538))^(1/5)\)

 

\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*20.2959972538*(1/5))\)

 

\(z^(1/5)≈2.003103242348*e^(i*4.059994507574)\)

 

\(z=r*(cos(\Theta)+i*sin(\Theta))\)

 

\(z≈2.003103242348*(cos(4.05994507574)+i*sin(4.05994507574))\)

 

\(z≈2.003103242348*(-0.60772245598411+i*-0.7941494925344)\)

 

\(z≈-1.2173308220295+i*1.5907634234047\)

 

\(z≈-1.2173308220295+1.5907634234047i\)

 

 

\(z^(1/5)≈(4\sqrt65*e^(i*26.5918256098))^(1/5)\)

 

\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*26.5918256098*(1/5))\)

 

\(z^(1/5)≈2.003103242348*e^(i*5.315836512196)\)

 

\(z=r*(cos(\Theta)+i*sin(\Theta))\)

 

\(z≈2.003103242348*(cos(5.315836512196)+i*sin(5.315836512196))\)

 

\(z≈2.003103242348*(0.56748448302716+i*-0.82338409112843)\)

 

\(z≈-1.1367300079339+i*-1.6493233426371\)

 

\(z≈-1.1367300079339+-1.6493233426371i\)

 

\(z≈-1.1367300079339-1.6493233426371i\)

gibsonj338 Nov 26, 2015

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