Answer
The five fifth roots are:
$96.59+25.88 i, \quad 5.23+99.86 i, \quad-93.36+35.84 i, \quad-62.93-77.71 i, \quad 54.46-83.87 i$
Work Step by Step
\[
=10^{10}\left[\cos 75^{\circ}+i \sin 75^{\circ}\right]=10^{10} \text { cis } 75^{\circ}
\]
Using the roots theorem:
\[
\left[\cos \left(\frac{\theta}{n}+\frac{360^{\circ}}{n} k\right)+i \sin \left(\frac{\theta}{n}+\frac{360^{\circ}}{n} k\right)\right]r^{\frac{1}{n}}=w_{k}
\]
When $0=k$ :
\[
\left(10^{10}\right)^{\frac{1}{3}}\left[\cos \left(\frac{75^{\circ}}{5}+\frac{360^{\circ}}{5}(0)\right)+i \sin \left(\frac{75^{\circ}}{5}+\frac{360^{\circ}}{5}(0)\right)\right]=w_{0}
\]
Simplify:
\[
=10^{2}\left[\cos 15^{\circ}+i \sin 15^{\circ}\right]
\]
Using a calculator:
\[
=[0.9659+0.2588 i]100
\]
Multiply:
\[
=96.59+25.88 i
\]
When $1=k$ :
\[
\left(10^{10}\right)^{\frac{1}{5}}\left[\cos \left(\frac{75^{\circ}}{5}+\frac{360^{\circ}}{5}(1)\right)+i \sin \left(\frac{75^{\circ}}{5}+\frac{360^{\circ}}{5}(1)\right)\right]=w_{1}
\]
Simplify:
\[
\begin{array}{c}
=10^{2}\left[\cos \left(15^{\circ}+72^{\circ}\right)+i \sin \left(15^{\circ}+72^{\circ}\right)\right] \\
=100\left[\cos 87^{\circ}+i \sin 87^{\circ}\right]
\end{array}
\]
Using a calculator:
\[
=1[0.0523+0.9986 i]100
\]
Multiply:
\[
=
\]
When $2=k$ :
\[
\left[\cos \left(\frac{75^{\circ}}{5}+\frac{360^{\circ}}{5}(2)\right)+i \sin \left(\frac{75^{\circ}}{5}+\frac{360^{\circ}}{5}(2)\right)\right]\left(10^{10}\right)^{t}=w_{2}
\]
Simplify:
\[
\begin{array}{c}
=10^{2}\left[\cos \left(15^{\circ}+144^{\circ}\right)+i \sin \left(15^{\circ}+144^{\circ}\right)\right] \\
=100\left[\cos 159^{\circ}+i \sin 159^{\circ}\right]
\end{array}
\]
Using a calculator:
\[
=1[-0.9336+0.3584 i]100
\]
Multiply:
When $3=k$ :
\[
\left(10^{10}\right)^{\frac{1}{5}}\left[\cos \left(\frac{75^{\circ}}{5}+\frac{360^{\circ}}{5}(3)\right)+i \sin \left(\frac{75^{\circ}}{5}+\frac{360^{\circ}}{5}(3)\right)\right]=w_{3}
\]
Simplify:
\[
\begin{array}{c}
=10^{2}\left[\cos \left(15^{\circ}+216^{\circ}\right)+i \sin \left(15^{\circ}+216^{\circ}\right)\right] \\
=100\left[\cos 231^{\circ}+i \sin 231^{\circ}\right]
\end{array}
\]
Using a calculator:
\[
=[-0.6293-0.7771 i]100
\]
Multiply:
When $4=k$ :
\[
\left(10^{10}\right)^{\frac{1}{5}}\left[\cos \left(\frac{75^{\circ}}{5}+\frac{360^{\circ}}{5}(4)\right)+i \sin \left(\frac{75^{\circ}}{5}+\frac{360^{\circ}}{5}(4)\right)\right]=w_{4}
\]
Simplify:
\[
\begin{array}{c}
=10^{2}\left[\cos \left(15^{\circ}+288^{\circ}\right)+i \sin \left(15^{\circ}+288^{\circ}\right)\right] \\
=10^{2}\left[\cos 303^{\circ}+i \sin 303^{\circ}\right]
\end{array}
\]
Using a calculator:
\[
=[0.5446-0.8387 i]100
\]
Multiply:
\[
=54.46-83.87 i
\]
