Linear Algebra Examples

$1 + i$

Step 1

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

$r = \sqrt{1^{2} + 1^{2}}$

Step 2

Simplify $\sqrt{1^{2} + 1^{2}}$ .

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Step 2.1

One to any power is one.

$r = \sqrt{1 + 1^{2}}$

Step 2.2

One to any power is one.

$r = \sqrt{1 + 1}$

Step 2.3

Add $1$ and $1$ .

$r = \sqrt{2}$

Step 3

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

$θ̂ = \arctan (| \frac{1}{1} |)$

Step 4

Simplify $\arctan (| \frac{1}{1} |)$ .

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Step 4.1

Divide $1$ by $1$ .

$θ̂ = \arctan (| 1 |)$

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Remove parentheses.

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Step 8.5.1

Remove parentheses.

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

Step 8.5.2

Remove parentheses.

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

Step 8.5.3

Remove parentheses.

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

Step 8.5.4

Remove parentheses.

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

Step 8.5.5

Remove parentheses.

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

Step 8.5.6

Remove parentheses.

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

Step 8.5.7

Remove parentheses.

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

Step 8.5.8

Remove parentheses.

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

Step 9

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

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Step 9.1

Remove parentheses.

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

Step 9.2

Multiply $2 π (0)$ .

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Step 9.2.1

Multiply $0$ by $2$ .

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

Step 9.2.2

Multiply $0$ by $π$ .

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

Step 9.3

Add $\frac{π}{4}$ and $0$ .

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

Step 9.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 9.5

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

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Step 9.5.1

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

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

Step 9.5.2

Multiply $4$ by $3$ .

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

Step 10

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

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Step 10.1

Remove parentheses.

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

Step 10.2

Multiply $2$ by $1$ .

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

Step 10.3

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

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

Step 10.4

Combine $2 π$ and $\frac{4}{4}$ .

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

Step 10.5

Combine the numerators over the common denominator.

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

Step 10.6

Simplify the numerator.

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Step 10.6.1

Multiply $4$ by $2$ .

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

Step 10.6.2

Add $π$ and $8 π$ .

$k = 1 : {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{9 π}{4}}{3})$

Step 10.7

Multiply the numerator by the reciprocal of the denominator.

$k = 1 : {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{9 π}{4} \cdot \frac{1}{3})$

Step 10.8

Cancel the common factor of $3$ .

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Step 10.8.1

Factor $3$ out of $9 π$ .

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

Step 10.8.2

Cancel the common factor.

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

Step 10.8.3

Rewrite the expression.

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

Step 11

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

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Step 11.1

Remove parentheses.

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

Step 11.2

Multiply $2$ by $2$ .

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

Step 11.3

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

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

Step 11.4

Combine $4 π$ and $\frac{4}{4}$ .

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

Step 11.5

Combine the numerators over the common denominator.

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

Step 11.6

Simplify the numerator.

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Step 11.6.1

Multiply $4$ by $4$ .

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

Step 11.6.2

Add $π$ and $16 π$ .

$k = 2 : {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{17 π}{4}}{3})$

Step 11.7

Multiply the numerator by the reciprocal of the denominator.

$k = 2 : {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{17 π}{4} \cdot \frac{1}{3})$

Step 11.8

Multiply $\frac{17 π}{4} \cdot \frac{1}{3}$ .

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Step 11.8.1

Multiply $\frac{17 π}{4}$ by $\frac{1}{3}$ .

$k = 2 : {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{17 π}{4 \cdot 3})$

Step 11.8.2

Multiply $4$ by $3$ .

$k = 2 : {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{17 π}{12})$

Step 12

List the solutions.

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

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

$k = 2 : {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{17 π}{12})$