Precalculus Examples

$- \sqrt{6} + \sqrt{2} i$

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

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

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

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

Substitute the actual values of $a = - \sqrt{6}$ and $b = \sqrt{2}$ .

$| z | = \sqrt{{(\sqrt{2})}^{2} + {(- \sqrt{6})}^{2}}$

Find $| z |$ .

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

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

$| z | = \sqrt{{(2^{\frac{1}{2}})}^{2} + {(- \sqrt{6})}^{2}}$

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

$| z | = \sqrt{2^{\frac{1}{2} \cdot 2} + {(- \sqrt{6})}^{2}}$

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

$| z | = \sqrt{2^{\frac{2}{2}} + {(- \sqrt{6})}^{2}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$| z | = \sqrt{2^{\frac{2}{2}} + {(- \sqrt{6})}^{2}}$

Rewrite the expression.

$| z | = \sqrt{2 + {(- \sqrt{6})}^{2}}$

Evaluate the exponent.

$| z | = \sqrt{2 + {(- \sqrt{6})}^{2}}$

Simplify the expression.

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Apply the product rule to $- \sqrt{6}$ .

$| z | = \sqrt{2 + {(- 1)}^{2} {\sqrt{6}}^{2}}$

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

$| z | = \sqrt{2 + 1 {\sqrt{6}}^{2}}$

Multiply ${\sqrt{6}}^{2}$ by $1$ .

$| z | = \sqrt{2 + {\sqrt{6}}^{2}}$

Rewrite ${\sqrt{6}}^{2}$ as $6$ .

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Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{6}$ as $6^{\frac{1}{2}}$ .

$| z | = \sqrt{2 + {(6^{\frac{1}{2}})}^{2}}$

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

$| z | = \sqrt{2 + 6^{\frac{1}{2} \cdot 2}}$

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

$| z | = \sqrt{2 + 6^{\frac{2}{2}}}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$| z | = \sqrt{2 + 6^{\frac{2}{2}}}$

Rewrite the expression.

$| z | = \sqrt{2 + 6}$

Evaluate the exponent.

$| z | = \sqrt{2 + 6}$

Add $2$ and $6$ .

$| z | = \sqrt{8}$

Rewrite $8$ as $2^{2} \cdot 2$ .

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Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

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

$θ = \arctan (\frac{\sqrt{2}}{- \sqrt{6}})$

Since inverse tangent of $\frac{\sqrt{2}}{- \sqrt{6}}$ produces an angle in the second quadrant, the value of the angle is $\frac{5 π}{6}$ .

$θ = \frac{5 π}{6}$

Substitute the values of $θ = \frac{5 π}{6}$ and $| z | = 2 \sqrt{2}$ .

$2 \sqrt{2} (\cos (\frac{5 π}{6}) + i \sin (\frac{5 π}{6}))$