Linear Algebra Examples

$- 3 + 4 i$

Step 1

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

$r = \sqrt{{(- 3)}^{2} + 4^{2}}$

Step 2

Simplify $\sqrt{{(- 3)}^{2} + 4^{2}}$ .

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Step 2.1

Raise $- 3$ to the power of $2$ .

$r = \sqrt{9 + 4^{2}}$

Step 2.2

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

$r = \sqrt{9 + 16}$

Step 2.3

Add $9$ and $16$ .

$r = \sqrt{25}$

Step 2.4

Rewrite $25$ as $5^{2}$ .

$r = \sqrt{5^{2}}$

Step 2.5

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

$r = 5$

Step 3

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

$θ̂ = \arctan (| \frac{4}{- 3} |)$

Step 4

Simplify $\arctan (| \frac{4}{- 3} |)$ .

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Step 4.1

Move the negative in front of the fraction.

$θ̂ = \arctan (| - \frac{4}{3} |)$

Step 4.2

$- \frac{4}{3}$ is approximately $- 1.\overline{3}$ which is negative so negate $- \frac{4}{3}$ and remove the absolute value

$θ̂ = \arctan (\frac{4}{3})$

Step 4.3

Evaluate $\arctan (\frac{4}{3})$ .

$θ̂ = 0.92729521$

Step 5

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

Quadrant $2$

Step 6

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

$θ = π - 0.92729521$

Step 7

Simplify $θ$ .

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Step 7.1

Replace with decimal approximation.

$3.14159265 - 0.92729521$

Step 7.2

Subtract $0.92729521$ from $3.14159265$ .

$2.21429743$

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

Replace with decimal approximation.

${(5)}^{\frac{1}{3}} c i s \frac{3.14159265 - 0.92729521 + 2 π k}{3}$

Step 9.2

Subtract $0.92729521$ from $3.14159265$ .

${(5)}^{\frac{1}{3}} c i s \frac{2.21429743 + 2 π k}{3}$

Step 9.3

Combine ${(5)}^{\frac{1}{3}}$ and $\frac{2.21429743 + 2 π k}{3}$ .

$c i s \frac{{(5)}^{\frac{1}{3}} (2.21429743 + 2 π k)}{3}$

Step 9.4

Combine $c$ and $\frac{{(5)}^{\frac{1}{3}} (2.21429743 + 2 π k)}{3}$ .

$i s \frac{c ({(5)}^{\frac{1}{3}} (2.21429743 + 2 π k))}{3}$

Step 9.5

Combine $i$ and $\frac{c ({(5)}^{\frac{1}{3}} (2.21429743 + 2 π k))}{3}$ .

$s \frac{i (c ({(5)}^{\frac{1}{3}} (2.21429743 + 2 π k)))}{3}$

Step 9.6

Combine $s$ and $\frac{i (c ({(5)}^{\frac{1}{3}} (2.21429743 + 2 π k)))}{3}$ .

$\frac{s (i (c ({(5)}^{\frac{1}{3}} (2.21429743 + 2 π k))))}{3}$

Step 9.7

Remove parentheses.

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Step 9.7.1

Remove parentheses.

$\frac{s (i (c (5^{\frac{1}{3}} (2.21429743 + 2 π k))))}{3}$

Step 9.7.2

Remove parentheses.

$\frac{s (i (c \cdot 5^{\frac{1}{3}} (2.21429743 + 2 π k)))}{3}$

Step 9.7.3

Remove parentheses.

$\frac{s (i (c \cdot 5^{\frac{1}{3}}) (2.21429743 + 2 π k))}{3}$

Step 9.7.4

Remove parentheses.

$\frac{s (i c \cdot 5^{\frac{1}{3}} (2.21429743 + 2 π k))}{3}$

Step 9.7.5

Remove parentheses.

$\frac{s (i c \cdot 5^{\frac{1}{3}}) (2.21429743 + 2 π k)}{3}$

Step 9.7.6

Remove parentheses.

$\frac{s (i c) \cdot 5^{\frac{1}{3}} (2.21429743 + 2 π k)}{3}$

Step 9.7.7

Remove parentheses.

$\frac{s i c \cdot 5^{\frac{1}{3}} (2.21429743 + 2 π k)}{3}$

Step 10

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

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Step 10.1

Remove parentheses.

$k = 0 : 5^{\frac{1}{3}} c i s (\frac{(π - 0.92729521) + 2 π (0)}{3})$

Step 10.2

Replace with decimal approximation.

$k = 0 : 5^{\frac{1}{3}} c i s (\frac{3.14159265 - 0.92729521 + 2 π (0)}{3})$

Step 10.3

Subtract $0.92729521$ from $3.14159265$ .

$k = 0 : 5^{\frac{1}{3}} c i s (\frac{2.21429743 + 2 π (0)}{3})$

Step 10.4

Multiply $2 π (0)$ .

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Step 10.4.1

Multiply $0$ by $2$ .

$k = 0 : 5^{\frac{1}{3}} c i s (\frac{2.21429743 + 0 π}{3})$

Step 10.4.2

Multiply $0$ by $π$ .

$k = 0 : 5^{\frac{1}{3}} c i s (\frac{2.21429743 + 0}{3})$

Step 10.5

Add $2.21429743$ and $0$ .

$k = 0 : 5^{\frac{1}{3}} c i s (\frac{2.21429743}{3})$

Step 10.6

Divide $2.21429743$ by $3$ .

$k = 0 : 5^{\frac{1}{3}} c i s \cdot 0.73809914$

Step 10.7

Multiply $5^{\frac{1}{3}} c i s$ by $0.73809914$ .

$k = 0 : 5^{\frac{1}{3}} c i s \cdot (0.73809914)$

Step 11

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

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Step 11.1

Remove parentheses.

$k = 1 : 5^{\frac{1}{3}} c i s (\frac{(π - 0.92729521) + 2 π (1)}{3})$

Step 11.2

Replace with decimal approximation.

$k = 1 : 5^{\frac{1}{3}} c i s (\frac{3.14159265 - 0.92729521 + 2 π (1)}{3})$

Step 11.3

Subtract $0.92729521$ from $3.14159265$ .

$k = 1 : 5^{\frac{1}{3}} c i s (\frac{2.21429743 + 2 π (1)}{3})$

Step 11.4

Multiply $2$ by $1$ .

$k = 1 : 5^{\frac{1}{3}} c i s (\frac{2.21429743 + 2 π}{3})$

Step 11.5

Add $2.21429743$ and $2 π$ .

$k = 1 : 5^{\frac{1}{3}} c i s (\frac{8.49748274}{3})$

Step 11.6

Divide $8.49748274$ by $3$ .

$k = 1 : 5^{\frac{1}{3}} c i s \cdot 2.83249424$

Step 11.7

Multiply $5^{\frac{1}{3}} c i s$ by $2.83249424$ .

$k = 1 : 5^{\frac{1}{3}} c i s \cdot (2.83249424)$

Step 12

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

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Step 12.1

Remove parentheses.

$k = 2 : 5^{\frac{1}{3}} c i s (\frac{(π - 0.92729521) + 2 π (2)}{3})$

Step 12.2

Replace with decimal approximation.

$k = 2 : 5^{\frac{1}{3}} c i s (\frac{3.14159265 - 0.92729521 + 2 π (2)}{3})$

Step 12.3

Subtract $0.92729521$ from $3.14159265$ .

$k = 2 : 5^{\frac{1}{3}} c i s (\frac{2.21429743 + 2 π (2)}{3})$

Step 12.4

Multiply $2$ by $2$ .

$k = 2 : 5^{\frac{1}{3}} c i s (\frac{2.21429743 + 4 π}{3})$

Step 12.5

Add $2.21429743$ and $4 π$ .

$k = 2 : 5^{\frac{1}{3}} c i s (\frac{14.78066804}{3})$

Step 12.6

Divide $14.78066804$ by $3$ .

$k = 2 : 5^{\frac{1}{3}} c i s \cdot 4.92688934$

Step 12.7

Multiply $5^{\frac{1}{3}} c i s$ by $4.92688934$ .

$k = 2 : 5^{\frac{1}{3}} c i s \cdot (4.92688934)$

Step 13

List the solutions.

$k = 0 : 5^{\frac{1}{3}} c i s \cdot (0.73809914)$

$k = 1 : 5^{\frac{1}{3}} c i s \cdot (2.83249424)$

$k = 2 : 5^{\frac{1}{3}} c i s \cdot (4.92688934)$