Linear Algebra Examples

4 i

$4 i$

Step 1

Calculate the distance from

(a, b)

$(a, b)$ to the origin using the formula

r = \sqrt{a^{2} + b^{2}}

$r = \sqrt{a^{2} + b^{2}}$ .

r = \sqrt{0^{2} + 4^{2}}

$r = \sqrt{0^{2} + 4^{2}}$

Step 2

Simplify

\sqrt{0^{2} + 4^{2}}

$\sqrt{0^{2} + 4^{2}}$ .

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Step 2.1

Raising

0

$0$ to any positive power yields

0

$0$ .

r = \sqrt{0 + 4^{2}}

$r = \sqrt{0 + 4^{2}}$

Step 2.2

Raise

4

$4$ to the power of

2

$2$ .

r = \sqrt{0 + 16}

$r = \sqrt{0 + 16}$

Step 2.3

Add

0

$0$ and

16

$16$ .

r = \sqrt{16}

$r = \sqrt{16}$

Step 2.4

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

r = \sqrt{4^{2}}

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

Step 2.5

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

r = 4

$r = 4$

r = 4

$r = 4$

Step 3

Calculate reference angle

θˆ = arctan (∣ ∣ ∣ \frac{b}{a} ∣ ∣ ∣)

$θ̂ = \arctan (| \frac{b}{a} |)$ .

θˆ = arctan (∣ ∣ ∣ \frac{4}{0} ∣ ∣ ∣)

$θ̂ = \arctan (| \frac{4}{0} |)$

Step 4

The equation has an undefined fraction.

Undefined

Step 5

Since the y-coordinate is positive and the x-coordinate is

0

$0$ , the point is located on y-axis between the first and fourth quadrants. The quadrants are labeled in counter-clockwise order, starting in the upper-right.

Between Quadrant

1

$1$ and

2

$2$

Step 6

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})

${(a + b i)}^{\frac{1}{n}} = r^{\frac{1}{n}} c i s (\frac{θ + 2 π k}{n})$ ,

k = 0, 1, \dots, n - 1

$k = 0, 1, \dots, n - 1$

Step 7

Substitute

r

$r$ ,

n

$n$ , and

θ

$θ$ into the formula.

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Step 7.1

Combine

{(4)}^{\frac{1}{2}}

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

\frac{θ + 2 π k}{2}

$\frac{θ + 2 π k}{2}$ .

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

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

Step 7.2

Combine

c

$c$ and

\frac{{(4)}^{\frac{1}{2}} (θ + 2 π k)}{2}

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

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

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

Step 7.3

Combine

i

$i$ and

\frac{c ({(4)}^{\frac{1}{2}} (θ + 2 π k))}{2}

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

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

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

Step 7.4

Combine

s

$s$ and

\frac{i (c ({(4)}^{\frac{1}{2}} (θ + 2 π k)))}{2}

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

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

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

Step 7.5

Remove parentheses.

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Step 7.5.1

Remove parentheses.

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

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

Step 7.5.2

Remove parentheses.

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

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

Step 7.5.3

Remove parentheses.

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

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

Step 7.5.4

Remove parentheses.

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

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

Step 7.5.5

Remove parentheses.

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

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

Step 7.5.6

Remove parentheses.

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

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

Step 7.5.7

Remove parentheses.

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

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

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

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

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

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

Step 8

Substitute

k = 0

$k = 0$ into the formula and simplify.

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Step 8.1

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 8.2

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

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

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

Step 8.3

Cancel the common factor of

2

$2$ .

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Step 8.3.1

Cancel the common factor.

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

Step 8.3.2

Rewrite the expression.

$k = 0 : 2 c i s (\frac{θ + 2 π (0)}{2})$

Step 8.4

Evaluate the exponent.

$k = 0 : 2 c i s (\frac{θ + 2 π (0)}{2})$

Step 8.5

Multiply

$2 π (0)$ .

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Step 8.5.1

Multiply

$0$ by

$2$ .

$k = 0 : 2 c i s (\frac{θ + 0 π}{2})$

Step 8.5.2

Multiply

$0$ by

$π$ .

$k = 0 : 2 c i s (\frac{θ + 0}{2})$

Step 9

Substitute

$k = 1$ into the formula and simplify.

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Step 9.1

Rewrite

$4$ as

$2^{2}$ .

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

Step 9.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

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

Step 9.3

Cancel the common factor of

$2$ .

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Step 9.3.1

Cancel the common factor.

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

Step 9.3.2

Rewrite the expression.

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

Step 9.4

Evaluate the exponent.

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

Step 9.5

Multiply

$2$ by

$1$ .

$k = 1 : 2 c i s (\frac{θ + 2 π}{2})$

Step 10

List the solutions.

$k = 0 : 2 c i s (\frac{θ + 0}{2})$

$k = 1 : 2 c i s (\frac{θ + 2 π}{2})$