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

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

Tap for more steps...

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

$4^{2} \cdot 2$ .

Tap for more steps...

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

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

Tap for more steps...

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 third quadrant because

$x$ and

$y$ are both negative. The quadrants are labeled in counter-clockwise order, starting in the upper-right.

Quadrant

$3$

Step 6

$(a, b)$ is in the third quadrant.

$θ = π + θ̂$

$θ = π + \frac{π}{4}$

Step 7

Simplify

$θ$ .

Tap for more steps...

Step 7.1

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

$π \cdot \frac{4}{4} + \frac{π}{4}$

Step 7.2

Combine fractions.

Tap for more steps...

Step 7.2.1

Combine

$π$ and

$\frac{4}{4}$ .

$\frac{π \cdot 4}{4} + \frac{π}{4}$

Step 7.2.2

Combine the numerators over the common denominator.

$\frac{π \cdot 4 + π}{4}$

Step 7.3

Simplify the numerator.

Tap for more steps...

Step 7.3.1

Move

$4$ to the left of

$π$ .

$\frac{4 \cdot π + π}{4}$

Step 7.3.2

Add

$4 π$ and

$π$ .

$\frac{5 π}{4}$

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.

Tap for more steps...

Step 9.1

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 9.2

Combine

$π$ and

$\frac{4}{4}$ .

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

Step 9.3

Combine the numerators over the common denominator.

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

Step 9.4

Add

$π \cdot 4$ and

$π$ .

Tap for more steps...

Step 9.4.1

Reorder

$π$ and

$4$ .

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

Step 9.4.2

Add

$4 \cdot π$ and

$π$ .

${(4 \sqrt{2})}^{\frac{1}{3}} c i s \frac{\frac{5 \cdot π}{4} + 2 π k}{3}$

Step 9.5

Combine

${(4 \sqrt{2})}^{\frac{1}{3}}$ and

$\frac{\frac{5 \cdot π}{4} + 2 π k}{3}$ .

$c i s \frac{{(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)}{3}$

Step 9.6

Combine

$c$ and

$\frac{{(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)}{3}$ .

$i s \frac{c ({(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k))}{3}$

Step 9.7

Combine

$i$ and

$\frac{c ({(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k))}{3}$ .

$s \frac{i (c ({(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)))}{3}$

Step 9.8

Combine

$s$ and

$\frac{i (c ({(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)))}{3}$ .

$\frac{s (i (c ({(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k))))}{3}$

Step 9.9

Remove parentheses.

Tap for more steps...

Step 9.9.1

Remove parentheses.

$\frac{s (i (c {(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)))}{3}$

Step 9.9.2

Remove parentheses.

$\frac{s (i (c {(4 \sqrt{2})}^{\frac{1}{3}}) (\frac{5 \cdot π}{4} + 2 π k))}{3}$

Step 9.9.3

Remove parentheses.

$\frac{s (i c {(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k))}{3}$

Step 9.9.4

Remove parentheses.

$\frac{s (i c {(4 \sqrt{2})}^{\frac{1}{3}}) (\frac{5 \cdot π}{4} + 2 π k)}{3}$

Step 9.9.5

Remove parentheses.

$\frac{s (i c) {(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)}{3}$

Step 9.9.6

Remove parentheses.

$\frac{s i c {(4 \sqrt{2})}^{\frac{1}{3}} (\frac{5 \cdot π}{4} + 2 π k)}{3}$

Step 10

Substitute

$k = 0$ into the formula and simplify.

Tap for more steps...

Step 10.1

Apply the product rule to

$4 \sqrt{2}$ .

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

Step 10.2

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 10.3

Combine

$π$ and

$\frac{4}{4}$ .

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

Step 10.4

Combine the numerators over the common denominator.

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

Step 10.5

Simplify the numerator.

Tap for more steps...

Step 10.5.1

Move

$4$ to the left of

$π$ .

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

Step 10.5.2

Add

$4 π$ and

$π$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + 2 π (0)}{3})$

Step 10.6

Multiply

$2 π (0)$ .

Tap for more steps...

Step 10.6.1

Multiply

$0$ by

$2$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + 0 π}{3})$

Step 10.6.2

Multiply

$0$ by

$π$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4} + 0}{3})$

Step 10.7

Add

$\frac{5 π}{4}$ and

$0$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π}{4}}{3})$

Step 10.8

Multiply the numerator by the reciprocal of the denominator.

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{5 π}{4} \cdot \frac{1}{3})$

Step 10.9

Multiply

$\frac{5 π}{4} \cdot \frac{1}{3}$ .

Tap for more steps...

Step 10.9.1

Multiply

$\frac{5 π}{4}$ by

$\frac{1}{3}$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{5 π}{4 \cdot 3})$

Step 10.9.2

Multiply

$4$ by

$3$ .

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{5 π}{12})$

Step 11

Substitute

$k = 1$ into the formula and simplify.

Tap for more steps...

Step 11.1

Apply the product rule to

$4 \sqrt{2}$ .

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

Step 11.2

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 11.3

Combine

$π$ and

$\frac{4}{4}$ .

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

Step 11.4

Combine the numerators over the common denominator.

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

Step 11.5

Simplify the numerator.

Tap for more steps...

Step 11.5.1

Move

$4$ to the left of

$π$ .

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

Step 11.5.2

Add

$4 π$ and

$π$ .

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

Step 11.6

Multiply

$2$ by

$1$ .

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

Step 11.7

To write

$2 π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 11.8

Combine

$2 π$ and

$\frac{4}{4}$ .

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

Step 11.9

Combine the numerators over the common denominator.

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

Step 11.10

Simplify the numerator.

Tap for more steps...

Step 11.10.1

Multiply

$4$ by

$2$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π + 8 π}{4}}{3})$

Step 11.10.2

Add

$5 π$ and

$8 π$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{13 π}{4}}{3})$

Step 11.11

Multiply the numerator by the reciprocal of the denominator.

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{13 π}{4} \cdot \frac{1}{3})$

Step 11.12

Multiply

$\frac{13 π}{4} \cdot \frac{1}{3}$ .

Tap for more steps...

Step 11.12.1

Multiply

$\frac{13 π}{4}$ by

$\frac{1}{3}$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{13 π}{4 \cdot 3})$

Step 11.12.2

Multiply

$4$ by

$3$ .

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{13 π}{12})$

Step 12

Substitute

$k = 2$ into the formula and simplify.

Tap for more steps...

Step 12.1

Apply the product rule to

$4 \sqrt{2}$ .

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

Step 12.2

To write

$π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 12.3

Combine

$π$ and

$\frac{4}{4}$ .

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

Step 12.4

Combine the numerators over the common denominator.

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

Step 12.5

Simplify the numerator.

Tap for more steps...

Step 12.5.1

Move

$4$ to the left of

$π$ .

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

Step 12.5.2

Add

$4 π$ and

$π$ .

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

Step 12.6

Multiply

$2$ by

$2$ .

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

Step 12.7

To write

$4 π$ as a fraction with a common denominator, multiply by

$\frac{4}{4}$ .

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

Step 12.8

Combine

$4 π$ and

$\frac{4}{4}$ .

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

Step 12.9

Combine the numerators over the common denominator.

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

Step 12.10

Simplify the numerator.

Tap for more steps...

Step 12.10.1

Multiply

$4$ by

$4$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{5 π + 16 π}{4}}{3})$

Step 12.10.2

Add

$5 π$ and

$16 π$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{\frac{21 π}{4}}{3})$

Step 12.11

Multiply the numerator by the reciprocal of the denominator.

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{21 π}{4} \cdot \frac{1}{3})$

Step 12.12

Cancel the common factor of

$3$ .

Tap for more steps...

Step 12.12.1

Factor

$3$ out of

$21 π$ .

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{3 (7 π)}{4} \cdot \frac{1}{3})$

Step 12.12.2

Cancel the common factor.

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{3 (7 π)}{4} \cdot \frac{1}{3})$

Step 12.12.3

Rewrite the expression.

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{7 π}{4})$

Step 13

List the solutions.

$k = 0 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{5 π}{12})$

$k = 1 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{13 π}{12})$

$k = 2 : 4^{\frac{1}{3}} {\sqrt{2}}^{\frac{1}{3}} c i s (\frac{7 π}{4})$