Linear Algebra Examples

$20 - 21 i$

Step 1

Calculate the distance from $(a, b)$ to the origin using the formula $r = \sqrt{a^{2} + b^{2}}$ .

$r = \sqrt{20^{2} + {(- 21)}^{2}}$

Step 2

Simplify $\sqrt{20^{2} + {(- 21)}^{2}}$ .

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Step 2.1

Raise $20$ to the power of $2$ .

$r = \sqrt{400 + {(- 21)}^{2}}$

Step 2.2

Raise $- 21$ to the power of $2$ .

$r = \sqrt{400 + 441}$

Step 2.3

Add $400$ and $441$ .

$r = \sqrt{841}$

Step 2.4

Rewrite $841$ as $29^{2}$ .

$r = \sqrt{29^{2}}$

Step 2.5

Pull terms out from under the radical, assuming positive real numbers.

$r = 29$

Step 3

Calculate reference angle $θ̂ = \arctan (| \frac{b}{a} |)$ .

$θ̂ = \arctan (| \frac{- 21}{20} |)$

Step 4

Simplify $\arctan (| \frac{- 21}{20} |)$ .

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Step 4.1

Move the negative in front of the fraction.

$θ̂ = \arctan (| - \frac{21}{20} |)$

Step 4.2

$- \frac{21}{20}$ is approximately $- 1.05$ which is negative so negate $- \frac{21}{20}$ and remove the absolute value

$θ̂ = \arctan (\frac{21}{20})$

Step 4.3

Evaluate $\arctan (\frac{21}{20})$ .

$θ̂ = 0.80978357$

Step 5

The point is located in the fourth quadrant because $x$ is positive and $y$ is negative. The quadrants are labeled in counter-clockwise order, starting in the upper-right.

Quadrant $4$

Step 6

$(a, b)$ is in the fourth quadrant. $θ = - θ̂$

$θ = - 0.80978357$

Step 7

Use the formula to find the roots of the complex number.

${(a + b i)}^{\frac{1}{n}} = r^{\frac{1}{n}} c i s (\frac{θ + 2 π k}{n})$ , $k = 0, 1, \dots, n - 1$

Step 8

Substitute $r$ , $n$ , and $θ$ into the formula.

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Step 8.1

Combine ${(29)}^{\frac{1}{2}}$ and $\frac{(- 0.80978357) + 2 π k}{2}$ .

$c i s \frac{{(29)}^{\frac{1}{2}} ((- 0.80978357) + 2 π k)}{2}$

Step 8.2

Combine $c$ and $\frac{{(29)}^{\frac{1}{2}} ((- 0.80978357) + 2 π k)}{2}$ .

$i s \frac{c ({(29)}^{\frac{1}{2}} ((- 0.80978357) + 2 π k))}{2}$

Step 8.3

Combine $i$ and $\frac{c ({(29)}^{\frac{1}{2}} ((- 0.80978357) + 2 π k))}{2}$ .

$s \frac{i (c ({(29)}^{\frac{1}{2}} ((- 0.80978357) + 2 π k)))}{2}$

Step 8.4

Combine $s$ and $\frac{i (c ({(29)}^{\frac{1}{2}} ((- 0.80978357) + 2 π k)))}{2}$ .

$\frac{s (i (c ({(29)}^{\frac{1}{2}} ((- 0.80978357) + 2 π k))))}{2}$

Step 8.5

Remove parentheses.

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Step 8.5.1

Remove parentheses.

$\frac{s (i (c (29^{\frac{1}{2}} ((- 0.80978357) + 2 π k))))}{2}$

Step 8.5.2

Remove parentheses.

$\frac{s (i (c (29^{\frac{1}{2}} (- 0.80978357 + 2 π k))))}{2}$

Step 8.5.3

Remove parentheses.

$\frac{s (i (c \cdot 29^{\frac{1}{2}} (- 0.80978357 + 2 π k)))}{2}$

Step 8.5.4

Remove parentheses.

$\frac{s (i (c \cdot 29^{\frac{1}{2}}) (- 0.80978357 + 2 π k))}{2}$

Step 8.5.5

Remove parentheses.

$\frac{s (i c \cdot 29^{\frac{1}{2}} (- 0.80978357 + 2 π k))}{2}$

Step 8.5.6

Remove parentheses.

$\frac{s (i c \cdot 29^{\frac{1}{2}}) (- 0.80978357 + 2 π k)}{2}$

Step 8.5.7

Remove parentheses.

$\frac{s (i c) \cdot 29^{\frac{1}{2}} (- 0.80978357 + 2 π k)}{2}$

Step 8.5.8

Remove parentheses.

$\frac{s i c \cdot 29^{\frac{1}{2}} (- 0.80978357 + 2 π k)}{2}$

Step 9

Substitute $k = 0$ into the formula and simplify.

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Step 9.1

Remove parentheses.

$k = 0 : 29^{\frac{1}{2}} c i s (\frac{(- 0.80978357) + 2 π (0)}{2})$

Step 9.2

Multiply $2 π (0)$ .

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Step 9.2.1

Multiply $0$ by $2$ .

$k = 0 : 29^{\frac{1}{2}} c i s (\frac{- 0.80978357 + 0 π}{2})$

Step 9.2.2

Multiply $0$ by $π$ .

$k = 0 : 29^{\frac{1}{2}} c i s (\frac{- 0.80978357 + 0}{2})$

Step 9.3

Add $- 0.80978357$ and $0$ .

$k = 0 : 29^{\frac{1}{2}} c i s (\frac{- 0.80978357}{2})$

Step 9.4

Divide $- 0.80978357$ by $2$ .

$k = 0 : 29^{\frac{1}{2}} c i s \cdot - 0.40489178$

Step 9.5

Multiply $29^{\frac{1}{2}} c i s$ by $- 0.40489178$ .

$k = 0 : 29^{\frac{1}{2}} c i s \cdot (- 0.40489178)$

Step 10

Substitute $k = 1$ into the formula and simplify.

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Step 10.1

Remove parentheses.

$k = 1 : 29^{\frac{1}{2}} c i s (\frac{(- 0.80978357) + 2 π (1)}{2})$

Step 10.2

Multiply $2$ by $1$ .

$k = 1 : 29^{\frac{1}{2}} c i s (\frac{- 0.80978357 + 2 π}{2})$

Step 10.3

Add $- 0.80978357$ and $2 π$ .

$k = 1 : 29^{\frac{1}{2}} c i s (\frac{5.47340173}{2})$

Step 10.4

Divide $5.47340173$ by $2$ .

$k = 1 : 29^{\frac{1}{2}} c i s \cdot 2.73670086$

Step 10.5

Multiply $29^{\frac{1}{2}} c i s$ by $2.73670086$ .

$k = 1 : 29^{\frac{1}{2}} c i s \cdot (2.73670086)$

Step 11

List the solutions.

$k = 0 : 29^{\frac{1}{2}} c i s \cdot (- 0.40489178)$

$k = 1 : 29^{\frac{1}{2}} c i s \cdot (2.73670086)$