Linear Algebra Examples

$- 3 + 3 \sqrt{3} i$

Step 1

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

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

Step 2

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

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Step 2.1

Simplify the expression.

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Step 2.1.1

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

$r = \sqrt{9 + {(3 \sqrt{3})}^{2}}$

Step 2.1.2

Apply the product rule to $3 \sqrt{3}$ .

$r = \sqrt{9 + 3^{2} {\sqrt{3}}^{2}}$

Step 2.1.3

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

$r = \sqrt{9 + 9 {\sqrt{3}}^{2}}$

Step 2.2

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$r = \sqrt{9 + 9 {(3^{\frac{1}{2}})}^{2}}$

Step 2.2.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$r = \sqrt{9 + 9 \cdot 3^{\frac{1}{2} \cdot 2}}$

Step 2.2.3

Combine $\frac{1}{2}$ and $2$ .

$r = \sqrt{9 + 9 \cdot 3^{\frac{2}{2}}}$

Step 2.2.4

Cancel the common factor of $2$ .

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Step 2.2.4.1

Cancel the common factor.

$r = \sqrt{9 + 9 \cdot 3^{\frac{2}{2}}}$

Step 2.2.4.2

Rewrite the expression.

$r = \sqrt{9 + 9 \cdot 3^{1}}$

Step 2.2.5

Evaluate the exponent.

$r = \sqrt{9 + 9 \cdot 3}$

Step 2.3

Simplify the expression.

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Step 2.3.1

Multiply $9$ by $3$ .

$r = \sqrt{9 + 27}$

Step 2.3.2

Add $9$ and $27$ .

$r = \sqrt{36}$

Step 2.3.3

Rewrite $36$ as $6^{2}$ .

$r = \sqrt{6^{2}}$

Step 2.3.4

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

$r = 6$

Step 3

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

$θ̂ = \arctan (| \frac{3 \sqrt{3}}{- 3} |)$

Step 4

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

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Step 4.1

Cancel the common factor of $3$ and $- 3$ .

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Step 4.1.1

Factor $3$ out of $3 \sqrt{3}$ .

$θ̂ = \arctan (| \frac{3 (\sqrt{3})}{- 3} |)$

Step 4.1.2

Move the negative one from the denominator of $\frac{\sqrt{3}}{- 1}$ .

$θ̂ = \arctan (| - 1 \cdot \sqrt{3} |)$

Step 4.2

Rewrite $- 1 \cdot \sqrt{3}$ as $- \sqrt{3}$ .

$θ̂ = \arctan (| - \sqrt{3} |)$

Step 4.3

$- \sqrt{3}$ is approximately $- 1.7320508$ which is negative so negate $- \sqrt{3}$ and remove the absolute value

$θ̂ = \arctan (\sqrt{3})$

Step 4.4

The exact value of $\arctan (\sqrt{3})$ is $\frac{π}{3}$ .

$θ̂ = \frac{π}{3}$

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. $θ = π - θ̂$

$θ = π - \frac{π}{3}$

Step 7

Simplify $θ$ .

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Step 7.1

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$π \cdot \frac{3}{3} - \frac{π}{3}$

Step 7.2

Combine fractions.

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Step 7.2.1

Combine $π$ and $\frac{3}{3}$ .

$\frac{π \cdot 3}{3} - \frac{π}{3}$

Step 7.2.2

Combine the numerators over the common denominator.

$\frac{π \cdot 3 - π}{3}$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Move $3$ to the left of $π$ .

$\frac{3 \cdot π - π}{3}$

Step 7.3.2

Subtract $π$ from $3 π$ .

$\frac{2 π}{3}$

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 $\frac{3}{3}$ .

${(6)}^{\frac{1}{2}} c i s \frac{π \cdot \frac{3}{3} - \frac{π}{3} + 2 π k}{2}$

Step 9.2

Combine $π$ and $\frac{3}{3}$ .

${(6)}^{\frac{1}{2}} c i s \frac{\frac{π \cdot 3}{3} - \frac{π}{3} + 2 π k}{2}$

Step 9.3

Combine the numerators over the common denominator.

${(6)}^{\frac{1}{2}} c i s \frac{\frac{π \cdot 3 - π}{3} + 2 π k}{2}$

Step 9.4

Subtract $π$ from $π \cdot 3$ .

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Step 9.4.1

Reorder $π$ and $3$ .

${(6)}^{\frac{1}{2}} c i s \frac{\frac{3 \cdot π - π}{3} + 2 π k}{2}$

Step 9.4.2

Subtract $π$ from $3 \cdot π$ .

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

Step 9.5

Combine ${(6)}^{\frac{1}{2}}$ and $\frac{\frac{2 \cdot π}{3} + 2 π k}{2}$ .

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

Step 9.6

Combine $c$ and $\frac{{(6)}^{\frac{1}{2}} (\frac{2 \cdot π}{3} + 2 π k)}{2}$ .

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

Step 9.7

Combine $i$ and $\frac{c ({(6)}^{\frac{1}{2}} (\frac{2 \cdot π}{3} + 2 π k))}{2}$ .

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

Step 9.8

Combine $s$ and $\frac{i (c ({(6)}^{\frac{1}{2}} (\frac{2 \cdot π}{3} + 2 π k)))}{2}$ .

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

Step 9.9

Remove parentheses.

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Step 9.9.1

Remove parentheses.

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

Step 9.9.2

Remove parentheses.

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

Step 9.9.3

Remove parentheses.

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

Step 9.9.4

Remove parentheses.

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

Step 9.9.5

Remove parentheses.

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

Step 9.9.6

Remove parentheses.

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

Step 9.9.7

Remove parentheses.

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

Step 10

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

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Step 10.1

Remove parentheses.

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{(π - \frac{π}{3}) + 2 π (0)}{2})$

Step 10.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{π \cdot \frac{3}{3} - \frac{π}{3} + 2 π (0)}{2})$

Step 10.3

Combine $π$ and $\frac{3}{3}$ .

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{\frac{π \cdot 3}{3} - \frac{π}{3} + 2 π (0)}{2})$

Step 10.4

Combine the numerators over the common denominator.

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{\frac{π \cdot 3 - π}{3} + 2 π (0)}{2})$

Step 10.5

Simplify the numerator.

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Step 10.5.1

Move $3$ to the left of $π$ .

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{\frac{3 \cdot π - π}{3} + 2 π (0)}{2})$

Step 10.5.2

Subtract $π$ from $3 π$ .

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{\frac{2 π}{3} + 2 π (0)}{2})$

Step 10.6

Multiply $2 π (0)$ .

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Step 10.6.1

Multiply $0$ by $2$ .

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{\frac{2 π}{3} + 0 π}{2})$

Step 10.6.2

Multiply $0$ by $π$ .

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{\frac{2 π}{3} + 0}{2})$

Step 10.7

Add $\frac{2 π}{3}$ and $0$ .

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{\frac{2 π}{3}}{2})$

Step 10.8

Multiply the numerator by the reciprocal of the denominator.

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{2 π}{3} \cdot \frac{1}{2})$

Step 10.9

Cancel the common factor of $2$ .

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Step 10.9.1

Factor $2$ out of $2 π$ .

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{2 (π)}{3} \cdot \frac{1}{2})$

Step 10.9.2

Cancel the common factor.

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{2 π}{3} \cdot \frac{1}{2})$

Step 10.9.3

Rewrite the expression.

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{π}{3})$

Step 11

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

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Step 11.1

Remove parentheses.

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

Step 11.2

To write $π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 11.3

Combine $π$ and $\frac{3}{3}$ .

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

Step 11.4

Combine the numerators over the common denominator.

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

Step 11.5

Simplify the numerator.

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Step 11.5.1

Move $3$ to the left of $π$ .

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

Step 11.5.2

Subtract $π$ from $3 π$ .

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

Step 11.6

Multiply $2$ by $1$ .

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

Step 11.7

To write $2 π$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 11.8

Combine $2 π$ and $\frac{3}{3}$ .

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

Step 11.9

Combine the numerators over the common denominator.

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

Step 11.10

Simplify the numerator.

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Step 11.10.1

Multiply $3$ by $2$ .

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

Step 11.10.2

Add $2 π$ and $6 π$ .

$k = 1 : 6^{\frac{1}{2}} c i s (\frac{\frac{8 π}{3}}{2})$

Step 11.11

Multiply the numerator by the reciprocal of the denominator.

$k = 1 : 6^{\frac{1}{2}} c i s (\frac{8 π}{3} \cdot \frac{1}{2})$

Step 11.12

Cancel the common factor of $2$ .

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Step 11.12.1

Factor $2$ out of $8 π$ .

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

Step 11.12.2

Cancel the common factor.

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

Step 11.12.3

Rewrite the expression.

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

Step 12

List the solutions.

$k = 0 : 6^{\frac{1}{2}} c i s (\frac{π}{3})$

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