Linear Algebra Examples

$5 + 2 i$

Step 1

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

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

Step 2

Simplify $\sqrt{5^{2} + 2^{2}}$ .

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Step 2.1

Raise $5$ to the power of $2$ .

$r = \sqrt{25 + 2^{2}}$

Step 2.2

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

$r = \sqrt{25 + 4}$

Step 2.3

Add $25$ and $4$ .

$r = \sqrt{29}$

Step 3

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

$θ̂ = \arctan (| \frac{2}{5} |)$

Step 4

Simplify $\arctan (| \frac{2}{5} |)$ .

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Step 4.1

$\frac{2}{5}$ is approximately $0.4$ which is positive so remove the absolute value

$θ̂ = \arctan (\frac{2}{5})$

Step 4.2

Evaluate $\arctan (\frac{2}{5})$ .

$θ̂ = 0.38050637$

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. $θ = θ̂$

$θ = 0.38050637$

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 ${(\sqrt{29})}^{\frac{1}{2}}$ and $\frac{(0.38050637) + 2 π k}{2}$ .

$c i s \frac{{(\sqrt{29})}^{\frac{1}{2}} ((0.38050637) + 2 π k)}{2}$

Step 8.2

Combine $c$ and $\frac{{(\sqrt{29})}^{\frac{1}{2}} ((0.38050637) + 2 π k)}{2}$ .

$i s \frac{c ({(\sqrt{29})}^{\frac{1}{2}} ((0.38050637) + 2 π k))}{2}$

Step 8.3

Combine $i$ and $\frac{c ({(\sqrt{29})}^{\frac{1}{2}} ((0.38050637) + 2 π k))}{2}$ .

$s \frac{i (c ({(\sqrt{29})}^{\frac{1}{2}} ((0.38050637) + 2 π k)))}{2}$

Step 8.4

Combine $s$ and $\frac{i (c ({(\sqrt{29})}^{\frac{1}{2}} ((0.38050637) + 2 π k)))}{2}$ .

$\frac{s (i (c ({(\sqrt{29})}^{\frac{1}{2}} ((0.38050637) + 2 π k))))}{2}$

Step 8.5

Remove parentheses.

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Step 8.5.1

Remove parentheses.

$\frac{s (i (c ({\sqrt{29}}^{\frac{1}{2}} ((0.38050637) + 2 π k))))}{2}$

Step 8.5.2

Remove parentheses.

$\frac{s (i (c ({\sqrt{29}}^{\frac{1}{2}} (0.38050637 + 2 π k))))}{2}$

Step 8.5.3

Remove parentheses.

$\frac{s (i (c {\sqrt{29}}^{\frac{1}{2}} (0.38050637 + 2 π k)))}{2}$

Step 8.5.4

Remove parentheses.

$\frac{s (i (c {\sqrt{29}}^{\frac{1}{2}}) (0.38050637 + 2 π k))}{2}$

Step 8.5.5

Remove parentheses.

$\frac{s (i c {\sqrt{29}}^{\frac{1}{2}} (0.38050637 + 2 π k))}{2}$

Step 8.5.6

Remove parentheses.

$\frac{s (i c {\sqrt{29}}^{\frac{1}{2}}) (0.38050637 + 2 π k)}{2}$

Step 8.5.7

Remove parentheses.

$\frac{s (i c) {\sqrt{29}}^{\frac{1}{2}} (0.38050637 + 2 π k)}{2}$

Step 8.5.8

Remove parentheses.

$\frac{s i c {\sqrt{29}}^{\frac{1}{2}} (0.38050637 + 2 π k)}{2}$

Step 9

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

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Step 9.1

Remove parentheses.

$k = 0 : {\sqrt{29}}^{\frac{1}{2}} c i s (\frac{(0.38050637) + 2 π (0)}{2})$

Step 9.2

Multiply $2 π (0)$ .

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Step 9.2.1

Multiply $0$ by $2$ .

$k = 0 : {\sqrt{29}}^{\frac{1}{2}} c i s (\frac{0.38050637 + 0 π}{2})$

Step 9.2.2

Multiply $0$ by $π$ .

$k = 0 : {\sqrt{29}}^{\frac{1}{2}} c i s (\frac{0.38050637 + 0}{2})$

Step 9.3

Add $0.38050637$ and $0$ .

$k = 0 : {\sqrt{29}}^{\frac{1}{2}} c i s (\frac{0.38050637}{2})$

Step 9.4

Divide $0.38050637$ by $2$ .

$k = 0 : {\sqrt{29}}^{\frac{1}{2}} c i s \cdot 0.19025318$

Step 9.5

Multiply ${\sqrt{29}}^{\frac{1}{2}} c i s$ by $0.19025318$ .

$k = 0 : {\sqrt{29}}^{\frac{1}{2}} c i s \cdot (0.19025318)$

Step 10

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

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Step 10.1

Remove parentheses.

$k = 1 : {\sqrt{29}}^{\frac{1}{2}} c i s (\frac{(0.38050637) + 2 π (1)}{2})$

Step 10.2

Multiply $2$ by $1$ .

$k = 1 : {\sqrt{29}}^{\frac{1}{2}} c i s (\frac{0.38050637 + 2 π}{2})$

Step 10.3

Add $0.38050637$ and $2 π$ .

$k = 1 : {\sqrt{29}}^{\frac{1}{2}} c i s (\frac{6.66369168}{2})$

Step 10.4

Divide $6.66369168$ by $2$ .

$k = 1 : {\sqrt{29}}^{\frac{1}{2}} c i s \cdot 3.33184584$

Step 10.5

Multiply ${\sqrt{29}}^{\frac{1}{2}} c i s$ by $3.33184584$ .

$k = 1 : {\sqrt{29}}^{\frac{1}{2}} c i s \cdot (3.33184584)$

Step 11

List the solutions.

$k = 0 : {\sqrt{29}}^{\frac{1}{2}} c i s \cdot (0.19025318)$

$k = 1 : {\sqrt{29}}^{\frac{1}{2}} c i s \cdot (3.33184584)$