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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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

Step 8.5

Remove parentheses.

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Step 8.5.1

Remove parentheses.

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

Step 8.5.2

Remove parentheses.

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

Step 8.5.3

Remove parentheses.

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

Step 8.5.4

Remove parentheses.

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

Step 8.5.5

Remove parentheses.

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

Step 8.5.6

Remove parentheses.

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

Step 8.5.7

Remove parentheses.

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Rewrite $4$ as $2^{2}$ .

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

Step 9.3

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

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

Step 9.4

Cancel the common factor of $2$ .

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Step 9.4.1

Cancel the common factor.

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

Step 9.4.2

Rewrite the expression.

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

Step 9.5

Evaluate the exponent.

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

Step 9.6

Multiply $2 π (0)$ .

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Step 9.6.1

Multiply $0$ by $2$ .

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

Step 9.6.2

Multiply $0$ by $π$ .

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

Step 9.7

Add $\frac{π}{4}$ and $0$ .

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

Step 9.8

Multiply the numerator by the reciprocal of the denominator.

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

Step 9.9

Multiply $\frac{π}{4} \cdot \frac{1}{2}$ .

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Step 9.9.1

Multiply $\frac{π}{4}$ by $\frac{1}{2}$ .

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

Step 9.9.2

Multiply $4$ by $2$ .

$k = 0 : 2 {\sqrt{2}}^{\frac{1}{2}} c i s (\frac{π}{8})$

Step 10

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

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Step 10.1

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

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

Step 10.2

Rewrite $4$ as $2^{2}$ .

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

Step 10.3

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

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

Step 10.4

Cancel the common factor of $2$ .

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Step 10.4.1

Cancel the common factor.

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

Step 10.4.2

Rewrite the expression.

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

Step 10.5

Evaluate the exponent.

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

Step 10.6

Multiply $2$ by $1$ .

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

Step 10.7

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

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

Step 10.8

Combine $2 π$ and $\frac{4}{4}$ .

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

Step 10.9

Combine the numerators over the common denominator.

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

Step 10.10

Simplify the numerator.

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Step 10.10.1

Multiply $4$ by $2$ .

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

Step 10.10.2

Add $π$ and $8 π$ .

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

Step 10.11

Multiply the numerator by the reciprocal of the denominator.

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

Step 10.12

Multiply $\frac{9 π}{4} \cdot \frac{1}{2}$ .

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Step 10.12.1

Multiply $\frac{9 π}{4}$ by $\frac{1}{2}$ .

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

Step 10.12.2

Multiply $4$ by $2$ .

$k = 1 : 2 {\sqrt{2}}^{\frac{1}{2}} c i s (\frac{9 π}{8})$

Step 11

List the solutions.

$k = 0 : 2 {\sqrt{2}}^{\frac{1}{2}} c i s (\frac{π}{8})$

$k = 1 : 2 {\sqrt{2}}^{\frac{1}{2}} c i s (\frac{9 π}{8})$