+1904 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.
+1904 \((4+32i)^(1/5)\)
\(r=\sqrt(a^2+b^2)\)
\(r=\sqrt(4^2+32^2)\)
\(r=\sqrt(16+1024)\)
\(r=\sqrt1040\)
\(r=4\sqrt65\)
\(tan(\Theta)=b/a\)
\(tan(\Theta)=32/4\)
\(tan(\Theta)=8\)
\(\Theta=tan^-1(8)\)
\(\Theta ≈1.4464413322481\)
\(z=r*e^(i*\Theta)\)
\(z≈4\sqrt65*e^(i*1.4464413322481)\)
\(z^(1/5)≈(4\sqrt65*e^(i*1.4464413322481))^(1/5)\)
\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*1.4464413322481*(1/5))\)
\(z^(1/5)≈2.003103242348*e^(i*1.0766143512748)\)
\(z=r*(cos(\Theta)+i*sin(\Theta))\)
\(z≈2.003103242348*(cos(1.0766143512748)+i*sin(1.0766143512748))\)
\(z≈2.003103242348*(0.4743116564868+i*0.88035700289064)\)
\(z≈0.9500952169545+i*1.763445966914\)
\(z≈0.9500952169545+1.763445966914i\)
\(z^(1/5)≈(4\sqrt65*e^(i*7.7296266394277))^(1/5)\)
\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*7.7296266394277*(1/5))\)
\(z^(1/5)≈2.003103242348*e^(i*1.5459253278855)\)
\(z=r*(cos(\Theta)+i*sin(\Theta))\)
\(z≈2.003103242348*(cos(1.5459253278855)+i*sin(1.5459253278855))\)
\(z≈2.003103242348*(0.024868434927217+i*0.99969073264899)\)
\(z≈-0.049814942634829+i*0.0001545372385216\)
\(z≈-0.049814942634829+0.0001545372385216i\)
\(z^(1/5)≈(4\sqrt65*e^(i*14.012811946607))^(1/5)\)
\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*14.012811946607*(1/5))\)
\(z^(1/5)≈2.003103242348*e^(i*1.6955283616309)\)
\(z=r*(cos(\Theta)+i*sin(\Theta))\)
\(z≈2.003103242348*(cos(1.6955283616309)+i*sin(1.6955283616309))\)
\(z≈2.003103242348*(-0.12440885450163+i*0.99223104009177)\)
\(z≈-0.24920377983349+i*1.9875412136019\)
\(z≈-0.24920377983349+1.9875412136019i\)
\(z^(1/5)≈(4\sqrt65*e^(i*20.2959972538))^(1/5)\)
\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*20.2959972538*(1/5))\)
\(z^(1/5)≈2.003103242348*e^(i*4.059994507574)\)
\(z=r*(cos(\Theta)+i*sin(\Theta))\)
\(z≈2.003103242348*(cos(4.05994507574)+i*sin(4.05994507574))\)
\(z≈2.003103242348*(-0.60772245598411+i*-0.7941494925344)\)
\(z≈-1.2173308220295+i*1.5907634234047\)
\(z≈-1.2173308220295+1.5907634234047i\)
\(z^(1/5)≈(4\sqrt65*e^(i*26.5918256098))^(1/5)\)
\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*26.5918256098*(1/5))\)
\(z^(1/5)≈2.003103242348*e^(i*5.315836512196)\)
\(z=r*(cos(\Theta)+i*sin(\Theta))\)
\(z≈2.003103242348*(cos(5.315836512196)+i*sin(5.315836512196))\)
\(z≈2.003103242348*(0.56748448302716+i*-0.82338409112843)\)
\(z≈-1.1367300079339+i*-1.6493233426371\)
\(z≈-1.1367300079339+-1.6493233426371i\)
\(z≈-1.1367300079339-1.6493233426371i\)
.
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
+1904 \((4+32i)^(1/5)\)
\(r=\sqrt(a^2+b^2)\)
\(r=\sqrt(4^2+32^2)\)
\(r=\sqrt(16+1024)\)
\(r=\sqrt1040\)
\(r=4\sqrt65\)
\(tan(\Theta)=b/a\)
\(tan(\Theta)=32/4\)
\(tan(\Theta)=8\)
\(\Theta=tan^-1(8)\)
\(\Theta ≈1.4464413322481\)
\(z=r*e^(i*\Theta)\)
\(z≈4\sqrt65*e^(i*1.4464413322481)\)
\(z^(1/5)≈(4\sqrt65*e^(i*1.4464413322481))^(1/5)\)
\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*1.4464413322481*(1/5))\)
\(z^(1/5)≈2.003103242348*e^(i*1.0766143512748)\)
\(z=r*(cos(\Theta)+i*sin(\Theta))\)
\(z≈2.003103242348*(cos(1.0766143512748)+i*sin(1.0766143512748))\)
\(z≈2.003103242348*(0.4743116564868+i*0.88035700289064)\)
\(z≈0.9500952169545+i*1.763445966914\)
\(z≈0.9500952169545+1.763445966914i\)
\(z^(1/5)≈(4\sqrt65*e^(i*7.7296266394277))^(1/5)\)
\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*7.7296266394277*(1/5))\)
\(z^(1/5)≈2.003103242348*e^(i*1.5459253278855)\)
\(z=r*(cos(\Theta)+i*sin(\Theta))\)
\(z≈2.003103242348*(cos(1.5459253278855)+i*sin(1.5459253278855))\)
\(z≈2.003103242348*(0.024868434927217+i*0.99969073264899)\)
\(z≈-0.049814942634829+i*0.0001545372385216\)
\(z≈-0.049814942634829+0.0001545372385216i\)
\(z^(1/5)≈(4\sqrt65*e^(i*14.012811946607))^(1/5)\)
\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*14.012811946607*(1/5))\)
\(z^(1/5)≈2.003103242348*e^(i*1.6955283616309)\)
\(z=r*(cos(\Theta)+i*sin(\Theta))\)
\(z≈2.003103242348*(cos(1.6955283616309)+i*sin(1.6955283616309))\)
\(z≈2.003103242348*(-0.12440885450163+i*0.99223104009177)\)
\(z≈-0.24920377983349+i*1.9875412136019\)
\(z≈-0.24920377983349+1.9875412136019i\)
\(z^(1/5)≈(4\sqrt65*e^(i*20.2959972538))^(1/5)\)
\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*20.2959972538*(1/5))\)
\(z^(1/5)≈2.003103242348*e^(i*4.059994507574)\)
\(z=r*(cos(\Theta)+i*sin(\Theta))\)
\(z≈2.003103242348*(cos(4.05994507574)+i*sin(4.05994507574))\)
\(z≈2.003103242348*(-0.60772245598411+i*-0.7941494925344)\)
\(z≈-1.2173308220295+i*1.5907634234047\)
\(z≈-1.2173308220295+1.5907634234047i\)
\(z^(1/5)≈(4\sqrt65*e^(i*26.5918256098))^(1/5)\)
\(z^(1/5)≈(4\sqrt65)^(1/5)*e^(i*26.5918256098*(1/5))\)
\(z^(1/5)≈2.003103242348*e^(i*5.315836512196)\)
\(z=r*(cos(\Theta)+i*sin(\Theta))\)
\(z≈2.003103242348*(cos(5.315836512196)+i*sin(5.315836512196))\)
\(z≈2.003103242348*(0.56748448302716+i*-0.82338409112843)\)
\(z≈-1.1367300079339+i*-1.6493233426371\)
\(z≈-1.1367300079339+-1.6493233426371i\)
\(z≈-1.1367300079339-1.6493233426371i\)
