Examples

Finding all 4th roots of 16

Compute all the $4^{\text{th}}$ roots of $16$.

Solution

Write $16=16e^{0i}$.

Evaluate roots formula:

$${\left(16e^{0i}\right)}^{\frac{1}{4}}\gg \gg w_k={16}^{\frac{1}{4}}{e}^{i(\frac{0}{4}+k\frac{2\pi }{4})}$$

Simplify:

$$\gg \gg 2e^{i\cdot k\frac{\pi }{2}}\gg \gg 2,2i,-2,-2i$$

Finding 2nd roots of 2i

Find both $2^{\text{nd}}$ roots of $2i$.

Solution

Write $2i=2e^{i\frac{\pi }{2}}$.

Evaluate roots formula:

$$\begin{array}{c}{\left(2e^{i\frac{\pi }{2}}\right)}^{\frac{1}{2}}\gg \gg w_k=\sqrt{2}{e}^{i(\frac{\pi /2}{2}+k\frac{2\pi }{2})}\\ \\ \gg \gg \sqrt{2}{e}^{i(\frac{\pi }{4}+k\pi )}\end{array}$$

Compute the options: $k=0,1$:

$$\gg \gg \sqrt{2}{e}^{i\frac{\pi }{4}},\sqrt{2}{e}^{i\frac{5\pi }{4}}$$

Convert to rectangular:

$$\begin{array}{c}\gg \gg \sqrt{2}(\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{2}}i),\sqrt{2}(-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}i)\\ \\ \gg \gg 1+i,1-i\end{array}$$

Some roots of unity

Find the $1^{\text{st}}$ and $2^{\text{nd}}$ and $3^{\text{rd}}$ and $4^{\text{th}}$ and $5^{\text{th}}$ and $6^{\text{th}}$ roots of the number $1$.

Solution

(1) $1^{\text{st}}$

Write $1=e^{0i}$. Evaluate roots formula. There is no possible $k$:

$${\left(e^{0i}\right)}^{\frac{1}{1}}\gg \gg e^{0i}\gg \gg 1$$

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(2) $2^{\text{nd}}$

Write $1=e^{0i}$. Evaluate roots formula in terms of $k$:

$${\left(e^{0i}\right)}^{\frac{1}{2}}\gg \gg w_k=e^{i(\frac{0}{2}+k\frac{2\pi }{2})}k=0,1$$

Compute the two options, $k=0,1$:

$$\gg \gg 1,e^{\pi i}\gg \gg 1,-1$$

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(3) $3^{\text{rd}}$

Evaluate roots formula in terms of $k$:

$${\left(e^{0i}\right)}^{\frac{1}{3}}\gg \gg w_k=e^{i(\frac{0}{3}+k\frac{2\pi }{3})}$$

Compute the options: $k=0,1,2$:

$$\gg \gg 1,e^{i\frac{2\pi }{3}},e^{i\frac{4\pi }{3}}\gg \gg 1,-\frac{1}{2}+\frac{\sqrt{3}}{2}i,-\frac{1}{2}-\frac{\sqrt{3}}{2}i$$

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(4) $4^{\text{th}}$

Evaluate roots formula:

$${\left(e^{0i}\right)}^{\frac{1}{4}}\gg \gg w_k=e^{i(\frac{0}{4}+k\frac{2\pi }{4})}$$

Compute the options: $k=0,1,2,3$:

$$1,e^{i\frac{\pi }{2}},e^{i\pi },e^{i\frac{3\pi }{2}}\gg \gg 1,i,-1,-i$$

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(5) $5^{\text{th}}$

Evaluate roots formula:

$${\left(e^{0i}\right)}^{\frac{1}{5}}\gg \gg w_k=e^{i(\frac{0}{5}+k\frac{2\pi }{5})}$$

Compute the options: $k=0,1,2,3,4$:

$$1,e^{i\frac{2\pi }{5}},e^{i\frac{4\pi }{5}},e^{i\frac{6\pi }{5}},e^{i\frac{8\pi }{5}}$$

Don’t simplify, it’s not feasible.

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(6) $6^{\text{th}}$

Evaluate roots formula:

$${\left(e^{0i}\right)}^{\frac{1}{6}}\gg \gg w_k=e^{i(\frac{0}{6}+k\frac{2\pi }{6})}$$

Compute the options: $k=0,1,2,3,4,5$:

$$1,e^{i\frac{2\pi }{6}},e^{i\frac{4\pi }{6}},e^{i\frac{6\pi }{6}},e^{i\frac{8\pi }{6}},e^{i\frac{10\pi }{6}}$$

Simplify:

$$\gg \gg 1,\frac{1}{2}+\frac{\sqrt{3}}{2}i,-\frac{1}{2}+\frac{\sqrt{3}}{2}i,-1,-\frac{1}{2}-\frac{\sqrt{3}}{2}i,\frac{1}{2}-\frac{\sqrt{3}}{2}i$$