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 first quadrant because $x$ and $y$ are both positive. The quadrants are labeled in counter-clockwise order, starting in the upper-right.

Quadrant $1$

Step 6

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

$θ = \frac{π}{4}$

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Remove parentheses.

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Step 8.5.1

Remove parentheses.

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

Step 8.5.2

Remove parentheses.

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

Step 8.5.3

Remove parentheses.

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

Step 8.5.4

Remove parentheses.

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

Step 8.5.5

Remove parentheses.

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

Step 8.5.6

Remove parentheses.

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

Step 8.5.7

Remove parentheses.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Multiply $2 π (0)$ .

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Step 9.2.1

Multiply $0$ by $2$ .

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

Step 9.2.2

Multiply $0$ by $π$ .

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

Step 9.3

Add $\frac{π}{4}$ and $0$ .

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

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 9.5

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

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Step 9.5.1

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

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

Step 9.5.2

Multiply $4$ by $4$ .

$k = 0 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{π}{16})$

Step 10

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

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Step 10.1

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

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

Step 10.2

Multiply $2$ by $1$ .

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

Step 10.3

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

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

Step 10.4

Combine $2 π$ and $\frac{4}{4}$ .

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

Step 10.5

Combine the numerators over the common denominator.

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

Step 10.6

Simplify the numerator.

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Step 10.6.1

Multiply $4$ by $2$ .

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

Step 10.6.2

Add $π$ and $8 π$ .

$k = 1 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{\frac{9 π}{4}}{4})$

Step 10.7

Multiply the numerator by the reciprocal of the denominator.

$k = 1 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{9 π}{4} \cdot \frac{1}{4})$

Step 10.8

Multiply $\frac{9 π}{4} \cdot \frac{1}{4}$ .

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Step 10.8.1

Multiply $\frac{9 π}{4}$ by $\frac{1}{4}$ .

$k = 1 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{9 π}{4 \cdot 4})$

Step 10.8.2

Multiply $4$ by $4$ .

$k = 1 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{9 π}{16})$

Step 11

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

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Step 11.1

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

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

Step 11.2

Multiply $2$ by $2$ .

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

Step 11.3

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

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

Step 11.4

Combine $4 π$ and $\frac{4}{4}$ .

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

Step 11.5

Combine the numerators over the common denominator.

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

Step 11.6

Simplify the numerator.

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Step 11.6.1

Multiply $4$ by $4$ .

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

Step 11.6.2

Add $π$ and $16 π$ .

$k = 2 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{\frac{17 π}{4}}{4})$

Step 11.7

Multiply the numerator by the reciprocal of the denominator.

$k = 2 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{17 π}{4} \cdot \frac{1}{4})$

Step 11.8

Multiply $\frac{17 π}{4} \cdot \frac{1}{4}$ .

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Step 11.8.1

Multiply $\frac{17 π}{4}$ by $\frac{1}{4}$ .

$k = 2 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{17 π}{4 \cdot 4})$

Step 11.8.2

Multiply $4$ by $4$ .

$k = 2 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{17 π}{16})$

Step 12

Substitute $k = 3$ into the formula and simplify.

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Step 12.1

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

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

Step 12.2

Multiply $3$ by $2$ .

$k = 3 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{\frac{π}{4} + 6 π}{4})$

Step 12.3

To write $6 π$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$k = 3 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{\frac{π}{4} + 6 π \cdot \frac{4}{4}}{4})$

Step 12.4

Combine $6 π$ and $\frac{4}{4}$ .

$k = 3 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{\frac{π}{4} + \frac{6 π \cdot 4}{4}}{4})$

Step 12.5

Combine the numerators over the common denominator.

$k = 3 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{\frac{π + 6 π \cdot 4}{4}}{4})$

Step 12.6

Simplify the numerator.

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Step 12.6.1

Multiply $4$ by $6$ .

$k = 3 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{\frac{π + 24 π}{4}}{4})$

Step 12.6.2

Add $π$ and $24 π$ .

$k = 3 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{\frac{25 π}{4}}{4})$

Step 12.7

Multiply the numerator by the reciprocal of the denominator.

$k = 3 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{25 π}{4} \cdot \frac{1}{4})$

Step 12.8

Multiply $\frac{25 π}{4} \cdot \frac{1}{4}$ .

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Step 12.8.1

Multiply $\frac{25 π}{4}$ by $\frac{1}{4}$ .

$k = 3 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{25 π}{4 \cdot 4})$

Step 12.8.2

Multiply $4$ by $4$ .

$k = 3 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{25 π}{16})$

Step 13

List the solutions.

$k = 0 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{π}{16})$

$k = 1 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{9 π}{16})$

$k = 2 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{17 π}{16})$

$k = 3 : 4^{\frac{1}{4}} {\sqrt{2}}^{\frac{1}{4}} c i s (\frac{25 π}{16})$