Algebra Examples

$- 4 - \sqrt{2} i$

Step 1

This is the trigonometric form of a complex number where $| z |$ is the modulus and $θ$ is the angle created on the complex plane.

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

Step 2

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

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

Step 3

Substitute the actual values of $a = - 4$ and $b = - 1 \sqrt{2}$ .

$| z | = \sqrt{{(- 1 \sqrt{2})}^{2} + {(- 4)}^{2}}$

Step 4

Find $| z |$ .

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Step 4.1

Simplify the expression.

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Step 4.1.1

Rewrite $- 1 \sqrt{2}$ as $- \sqrt{2}$ .

$| z | = \sqrt{{(- \sqrt{2})}^{2} + {(- 4)}^{2}}$

Step 4.1.2

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

$| z | = \sqrt{{(- 1)}^{2} {\sqrt{2}}^{2} + {(- 4)}^{2}}$

Step 4.1.3

Raise $- 1$ to the power of $2$ .

$| z | = \sqrt{1 {\sqrt{2}}^{2} + {(- 4)}^{2}}$

Step 4.1.4

Multiply ${\sqrt{2}}^{2}$ by $1$ .

$| z | = \sqrt{{\sqrt{2}}^{2} + {(- 4)}^{2}}$

Step 4.2

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

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Step 4.2.1

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

$| z | = \sqrt{{(2^{\frac{1}{2}})}^{2} + {(- 4)}^{2}}$

Step 4.2.2

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

$| z | = \sqrt{2^{\frac{1}{2} \cdot 2} + {(- 4)}^{2}}$

Step 4.2.3

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

$| z | = \sqrt{2^{\frac{2}{2}} + {(- 4)}^{2}}$

Step 4.2.4

Cancel the common factor of $2$ .

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Step 4.2.4.1

Cancel the common factor.

$| z | = \sqrt{2^{\frac{2}{2}} + {(- 4)}^{2}}$

Step 4.2.4.2

Rewrite the expression.

$| z | = \sqrt{2 + {(- 4)}^{2}}$

Step 4.2.5

Evaluate the exponent.

$| z | = \sqrt{2 + {(- 4)}^{2}}$

Step 4.3

Simplify the expression.

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Step 4.3.1

Raise $- 4$ to the power of $2$ .

$| z | = \sqrt{2 + 16}$

Step 4.3.2

Add $2$ and $16$ .

$| z | = \sqrt{18}$

Step 4.4

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 4.4.1

Factor $9$ out of $18$ .

$| z | = \sqrt{9 (2)}$

Step 4.4.2

Rewrite $9$ as $3^{2}$ .

$| z | = \sqrt{3^{2} \cdot 2}$

Step 4.5

Pull terms out from under the radical.

$| z | = 3 \sqrt{2}$

Step 5

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

$θ = \arctan (\frac{- 1 \sqrt{2}}{- 4})$

Step 6

Since inverse tangent of $\frac{- 1 \sqrt{2}}{- 4}$ produces an angle in the third quadrant, the value of the angle is $3.48142956$ .

$θ = 3.48142956$

Step 7

Substitute the values of $θ = 3.48142956$ and $| z | = 3 \sqrt{2}$ .

$3 \sqrt{2} (\cos (3.48142956) + i \sin (3.48142956))$