Linear Algebra Examples

$- 4 - 4 i$

Step 1

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

$r = \sqrt{{(- 4)}^{2} + {(- 4)}^{2}}$

Step 2

Simplify $\sqrt{{(- 4)}^{2} + {(- 4)}^{2}}$ .

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Step 2.1

Raise $- 4$ to the power of $2$ .

$r = \sqrt{16 + {(- 4)}^{2}}$

Step 2.2

Raise $- 4$ to the power of $2$ .

$r = \sqrt{16 + 16}$

Step 2.3

Add $16$ and $16$ .

$r = \sqrt{32}$

Step 2.4

Rewrite $32$ as $4^{2} \cdot 2$ .

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Step 2.4.1

Factor $16$ out of $32$ .

$r = \sqrt{16 (2)}$

Step 2.4.2

Rewrite $16$ as $4^{2}$ .

$r = \sqrt{4^{2} \cdot 2}$

Step 2.5

Pull terms out from under the radical.

$r = 4 \sqrt{2}$

Step 3

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

$θ̂ = \arctan (| \frac{- 4}{- 4} |)$

Step 4

Simplify $\arctan (| \frac{- 4}{- 4} |)$ .

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Step 4.1

Divide $- 4$ by $- 4$ .

$θ̂ = \arctan (| 1 |)$

Step 4.2

The absolute value is the distance between a number and zero. The distance between $0$ and $1$ is $1$ .

$θ̂ = \arctan (1)$

Step 4.3

The exact value of $\arctan (1)$ is $\frac{π}{4}$ .

$θ̂ = \frac{π}{4}$

Step 5

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

Quadrant $3$

Step 6

$(a, b)$ is in the third quadrant. $θ = π + θ̂$

$θ = π + \frac{π}{4}$

Step 7

Simplify $θ$ .

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Step 7.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$π \cdot \frac{4}{4} + \frac{π}{4}$

Step 7.2

Combine fractions.

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Step 7.2.1

Combine $π$ and $\frac{4}{4}$ .

$\frac{π \cdot 4}{4} + \frac{π}{4}$

Step 7.2.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 + π}{4}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Move $4$ to the left of $π$ .

$\frac{4 \cdot π + π}{4}$

Step 7.3.2

Add $4 π$ and $π$ .

$\frac{5 π}{4}$

Step 8

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 9

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

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Step 9.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

${(4 \sqrt{2})}^{\frac{1}{3}} c i s \frac{π \cdot \frac{4}{4} + \frac{π}{4} + 2 π k}{3}$

Step 9.2

Combine $π$ and $\frac{4}{4}$ .

${(4 \sqrt{2})}^{\frac{1}{3}} c i s \frac{\frac{π \cdot 4}{4} + \frac{π}{4} + 2 π k}{3}$

Step 9.3

Combine the numerators over the common denominator.

${(4 \sqrt{2})}^{\frac{1}{3}} c i s \frac{\frac{π \cdot 4 + π}{4} + 2 π k}{3}$

Step 9.4

Add $π \cdot 4$ and $π$ .

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Step 9.4.1

Reorder $π$ and $4$ .

${(4 \sqrt{2})}^{\frac{1}{3}} c i s \frac{\frac{4 \cdot π + π}{4} + 2 π k}{3}$

Step 9.4.2

Add $4 \cdot π$ and $π$ .

${(4 \sqrt{2})}^{\frac{1}{3}} c i s \frac{\frac{5 \cdot π}{4} + 2 π k}{3}$

Step 9.5

Combine ${(4 \sqrt{2})}^{\frac{1}{3}}$ and $\frac{\frac{5 \cdot π}{4} + 2 π k}{3}$ .

$c i s \frac{{(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)}{3}$

Step 9.6

Combine $c$ and $\frac{{(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)}{3}$ .

$i s \frac{c ({(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k))}{3}$

Step 9.7

Combine $i$ and $\frac{c ({(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k))}{3}$ .

$s \frac{i (c ({(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)))}{3}$

Step 9.8

Combine $s$ and $\frac{i (c ({(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)))}{3}$ .

$\frac{s (i (c ({(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k))))}{3}$

Step 9.9

Remove parentheses.

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Step 9.9.1

Remove parentheses.

$\frac{s (i (c {(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)))}{3}$

Step 9.9.2

Remove parentheses.

$\frac{s (i (c {(4 \sqrt{2})}^{\frac{1}{3}}) (\frac{5 \cdot π}{4} + 2 π k))}{3}$

Step 9.9.3

Remove parentheses.

$\frac{s (i c {(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k))}{3}$

Step 9.9.4

Remove parentheses.

$\frac{s (i c {(4 \sqrt{2})}^{\frac{1}{3}}) (\frac{5 \cdot π}{4} + 2 π k)}{3}$

Step 9.9.5

Remove parentheses.

$\frac{s (i c) {(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)}{3}$

Step 9.9.6

Remove parentheses.

$\frac{s i c {(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)}{3}$

Step 10

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

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Step 10.1

Apply the product rule to $4 \sqrt{2}$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{(π + \frac{π}{4}) + 2 π (0)}{3})$

Step 10.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{π \cdot \frac{4}{4} + \frac{π}{4} + 2 π (0)}{3})$

Step 10.3

Combine $π$ and $\frac{4}{4}$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{π \cdot 4}{4} + \frac{π}{4} + 2 π (0)}{3})$

Step 10.4

Combine the numerators over the common denominator.

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{π \cdot 4 + π}{4} + 2 π (0)}{3})$

Step 10.5

Simplify the numerator.

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Step 10.5.1

Move $4$ to the left of $π$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{4 \cdot π + π}{4} + 2 π (0)}{3})$

Step 10.5.2

Add $4 π$ and $π$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + 2 π (0)}{3})$

Step 10.6

Multiply $2 π (0)$ .

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Step 10.6.1

Multiply $0$ by $2$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + 0 π}{3})$

Step 10.6.2

Multiply $0$ by $π$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + 0}{3})$

Step 10.7

Add $\frac{5 π}{4}$ and $0$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4}}{3})$

Step 10.8

Multiply the numerator by the reciprocal of the denominator.

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{5 π}{4} \cdot \frac{1}{3})$

Step 10.9

Multiply $\frac{5 π}{4} \cdot \frac{1}{3}$ .

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Step 10.9.1

Multiply $\frac{5 π}{4}$ by $\frac{1}{3}$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{5 π}{4 \cdot 3})$

Step 10.9.2

Multiply $4$ by $3$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{5 π}{12})$

Step 11

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

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Step 11.1

Apply the product rule to $4 \sqrt{2}$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{(π + \frac{π}{4}) + 2 π (1)}{3})$

Step 11.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{π \cdot \frac{4}{4} + \frac{π}{4} + 2 π (1)}{3})$

Step 11.3

Combine $π$ and $\frac{4}{4}$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{π \cdot 4}{4} + \frac{π}{4} + 2 π (1)}{3})$

Step 11.4

Combine the numerators over the common denominator.

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{π \cdot 4 + π}{4} + 2 π (1)}{3})$

Step 11.5

Simplify the numerator.

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Step 11.5.1

Move $4$ to the left of $π$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{4 \cdot π + π}{4} + 2 π (1)}{3})$

Step 11.5.2

Add $4 π$ and $π$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + 2 π (1)}{3})$

Step 11.6

Multiply $2$ by $1$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + 2 π}{3})$

Step 11.7

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + 2 π \cdot \frac{4}{4}}{3})$

Step 11.8

Combine $2 π$ and $\frac{4}{4}$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + \frac{2 π \cdot 4}{4}}{3})$

Step 11.9

Combine the numerators over the common denominator.

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π + 2 π \cdot 4}{4}}{3})$

Step 11.10

Simplify the numerator.

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Step 11.10.1

Multiply $4$ by $2$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π + 8 π}{4}}{3})$

Step 11.10.2

Add $5 π$ and $8 π$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{13 π}{4}}{3})$

Step 11.11

Multiply the numerator by the reciprocal of the denominator.

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{13 π}{4} \cdot \frac{1}{3})$

Step 11.12

Multiply $\frac{13 π}{4} \cdot \frac{1}{3}$ .

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Step 11.12.1

Multiply $\frac{13 π}{4}$ by $\frac{1}{3}$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{13 π}{4 \cdot 3})$

Step 11.12.2

Multiply $4$ by $3$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{13 π}{12})$

Step 12

Substitute $k = 2$ into the formula and simplify.

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Step 12.1

Apply the product rule to $4 \sqrt{2}$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{(π + \frac{π}{4}) + 2 π (2)}{3})$

Step 12.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{π \cdot \frac{4}{4} + \frac{π}{4} + 2 π (2)}{3})$

Step 12.3

Combine $π$ and $\frac{4}{4}$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{π \cdot 4}{4} + \frac{π}{4} + 2 π (2)}{3})$

Step 12.4

Combine the numerators over the common denominator.

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{π \cdot 4 + π}{4} + 2 π (2)}{3})$

Step 12.5

Simplify the numerator.

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Step 12.5.1

Move $4$ to the left of $π$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{4 \cdot π + π}{4} + 2 π (2)}{3})$

Step 12.5.2

Add $4 π$ and $π$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + 2 π (2)}{3})$

Step 12.6

Multiply $2$ by $2$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + 4 π}{3})$

Step 12.7

To write $4 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + 4 π \cdot \frac{4}{4}}{3})$

Step 12.8

Combine $4 π$ and $\frac{4}{4}$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + \frac{4 π \cdot 4}{4}}{3})$

Step 12.9

Combine the numerators over the common denominator.

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π + 4 π \cdot 4}{4}}{3})$

Step 12.10

Simplify the numerator.

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Step 12.10.1

Multiply $4$ by $4$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π + 16 π}{4}}{3})$

Step 12.10.2

Add $5 π$ and $16 π$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{21 π}{4}}{3})$

Step 12.11

Multiply the numerator by the reciprocal of the denominator.

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{21 π}{4} \cdot \frac{1}{3})$

Step 12.12

Cancel the common factor of $3$ .

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Step 12.12.1

Factor $3$ out of $21 π$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{3 (7 π)}{4} \cdot \frac{1}{3})$

Step 12.12.2

Cancel the common factor.

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{3 (7 π)}{4} \cdot \frac{1}{3})$

Step 12.12.3

Rewrite the expression.

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{7 π}{4})$

Step 13

List the solutions.

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{5 π}{12})$

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{13 π}{12})$

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{7 π}{4})$