Linear Algebra Examples

| - 7 - 9 i |

$| - 7 - 9 i |$

Step 1

Use the formula

| a + b i | = \sqrt{a^{2} + b^{2}}

$| a + b i | = \sqrt{a^{2} + b^{2}}$ to find the magnitude.

\sqrt{{(- 7)}^{2} + {(- 9)}^{2}}

$\sqrt{{(- 7)}^{2} + {(- 9)}^{2}}$

Step 2

Raise

- 7

$- 7$ to the power of

2

$2$ .

\sqrt{49 + {(- 9)}^{2}}

$\sqrt{49 + {(- 9)}^{2}}$

Step 3

Raise

- 9

$- 9$ to the power of

2

$2$ .

\sqrt{49 + 81}

$\sqrt{49 + 81}$

Step 4

Add

49

$49$ and

81

$81$ .

\sqrt{130}

$\sqrt{130}$

Step 5

This is the trigonometric form of a complex number where

| z |

$| z |$ is the modulus and

θ

$θ$ is the angle created on the complex plane.

z = a + b i = | z | (\cos (θ) + i \sin (θ))

$z = a + b i = | z | (\cos (θ) + i \sin (θ))$

Step 6

The modulus of a complex number is the distance from the origin on the complex plane.

| z | = \sqrt{a^{2} + b^{2}}

$| z | = \sqrt{a^{2} + b^{2}}$ where

z = a + b i

$z = a + b i$

Step 7

Substitute the actual values of

a = \sqrt{130}

$a = \sqrt{130}$ and

b = 0

$b = 0$ .

| z | = \sqrt{0^{2} + {(\sqrt{130})}^{2}}

$| z | = \sqrt{0^{2} + {(\sqrt{130})}^{2}}$

Step 8

Find

| z |

$| z |$ .

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Step 8.1

Raising

0

$0$ to any positive power yields

0

$0$ .

| z | = \sqrt{0 + {(\sqrt{130})}^{2}}

$| z | = \sqrt{0 + {(\sqrt{130})}^{2}}$

Step 8.2

Rewrite

{\sqrt{130}}^{2}

${\sqrt{130}}^{2}$ as

130

$130$ .

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Step 8.2.1

Use

\sqrt[n]{a^{x}} = a^{\frac{x}{n}}

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

\sqrt{130}

$\sqrt{130}$ as

130^{\frac{1}{2}}

$130^{\frac{1}{2}}$ .

| z | = \sqrt{0 + {(130^{\frac{1}{2}})}^{2}}

$| z | = \sqrt{0 + {(130^{\frac{1}{2}})}^{2}}$

Step 8.2.2

Apply the power rule and multiply exponents,

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

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

| z | = \sqrt{0 + 130^{\frac{1}{2} \cdot 2}}

$| z | = \sqrt{0 + 130^{\frac{1}{2} \cdot 2}}$

Step 8.2.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

| z | = \sqrt{0 + 130^{\frac{2}{2}}}

$| z | = \sqrt{0 + 130^{\frac{2}{2}}}$

Step 8.2.4

Cancel the common factor of

2

$2$ .

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Step 8.2.4.1

Cancel the common factor.

| z | = \sqrt{0 + 130^{\frac{2}{2}}}

$| z | = \sqrt{0 + 130^{\frac{2}{2}}}$

Step 8.2.4.2

Rewrite the expression.

| z | = \sqrt{0 + 130}

$| z | = \sqrt{0 + 130}$

| z | = \sqrt{0 + 130}

$| z | = \sqrt{0 + 130}$

Step 8.2.5

Evaluate the exponent.

| z | = \sqrt{0 + 130}

$| z | = \sqrt{0 + 130}$

| z | = \sqrt{0 + 130}

$| z | = \sqrt{0 + 130}$

Step 8.3

Add

0

$0$ and

130

$130$ .

| z | = \sqrt{130}

$| z | = \sqrt{130}$

| z | = \sqrt{130}

$| z | = \sqrt{130}$

Step 9

The angle of the point on the complex plane is the inverse tangent of the complex portion over the real portion.

θ = \arctan (\frac{0}{\sqrt{130}})

$θ = \arctan (\frac{0}{\sqrt{130}})$

Step 10

Since inverse tangent of

\frac{0}{\sqrt{130}}

$\frac{0}{\sqrt{130}}$ produces an angle in the first quadrant, the value of the angle is

0

$0$ .

θ = 0

$θ = 0$

Step 11

Substitute the values of

θ = 0

$θ = 0$ and

| z | = \sqrt{130}

$| z | = \sqrt{130}$ .

\sqrt{130} (\cos (0) + i \sin (0))

$\sqrt{130} (\cos (0) + i \sin (0))$