Precalculus Examples

$- 1 - \frac{\sqrt{3}}{3} \cdot i$

Step 1

Combine $i$ and $\frac{\sqrt{3}}{3}$ .

$- 1 - \frac{i \sqrt{3}}{3}$

Step 2

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 3

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 4

Substitute the actual values of $a = - 1$ and $b = - \frac{1 \sqrt{3}}{3}$ .

$| z | = \sqrt{{(- \frac{1 \sqrt{3}}{3})}^{2} + {(- 1)}^{2}}$

Step 5

Find $| z |$ .

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Step 5.1

Multiply $\sqrt{3}$ by $1$ .

$| z | = \sqrt{{(- \frac{\sqrt{3}}{3})}^{2} + {(- 1)}^{2}}$

Step 5.2

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.2.1

Apply the product rule to $- \frac{\sqrt{3}}{3}$ .

$| z | = \sqrt{{(- 1)}^{2} {(\frac{\sqrt{3}}{3})}^{2} + {(- 1)}^{2}}$

Step 5.2.2

Apply the product rule to $\frac{\sqrt{3}}{3}$ .

$| z | = \sqrt{{(- 1)}^{2} (\frac{{\sqrt{3}}^{2}}{3^{2}}) + {(- 1)}^{2}}$

Step 5.3

Simplify the expression.

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Step 5.3.1

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

$| z | = \sqrt{1 (\frac{{\sqrt{3}}^{2}}{3^{2}}) + {(- 1)}^{2}}$

Step 5.3.2

Multiply $\frac{{\sqrt{3}}^{2}}{3^{2}}$ by $1$ .

$| z | = \sqrt{\frac{{\sqrt{3}}^{2}}{3^{2}} + {(- 1)}^{2}}$

Step 5.4

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.4.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$| z | = \sqrt{\frac{{(3^{\frac{1}{2}})}^{2}}{3^{2}} + {(- 1)}^{2}}$

Step 5.4.2

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

$| z | = \sqrt{\frac{3^{\frac{1}{2} \cdot 2}}{3^{2}} + {(- 1)}^{2}}$

Step 5.4.3

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

$| z | = \sqrt{\frac{3^{\frac{2}{2}}}{3^{2}} + {(- 1)}^{2}}$

Step 5.4.4

Cancel the common factor of $2$ .

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Step 5.4.4.1

Cancel the common factor.

$| z | = \sqrt{\frac{3^{\frac{2}{2}}}{3^{2}} + {(- 1)}^{2}}$

Step 5.4.4.2

Rewrite the expression.

$| z | = \sqrt{\frac{3}{3^{2}} + {(- 1)}^{2}}$

Step 5.4.5

Evaluate the exponent.

$| z | = \sqrt{\frac{3}{3^{2}} + {(- 1)}^{2}}$

Step 5.5

Raise $3$ to the power of $2$ .

$| z | = \sqrt{\frac{3}{9} + {(- 1)}^{2}}$

Step 5.6

Cancel the common factor of $3$ and $9$ .

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Step 5.6.1

Factor $3$ out of $3$ .

$| z | = \sqrt{\frac{3 (1)}{9} + {(- 1)}^{2}}$

Step 5.6.2

Cancel the common factors.

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Step 5.6.2.1

Factor $3$ out of $9$ .

$| z | = \sqrt{\frac{3 \cdot 1}{3 \cdot 3} + {(- 1)}^{2}}$

Step 5.6.2.2

Cancel the common factor.

$| z | = \sqrt{\frac{3 \cdot 1}{3 \cdot 3} + {(- 1)}^{2}}$

Step 5.6.2.3

Rewrite the expression.

$| z | = \sqrt{\frac{1}{3} + {(- 1)}^{2}}$

Step 5.7

Simplify the expression.

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Step 5.7.1

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

$| z | = \sqrt{\frac{1}{3} + 1}$

Step 5.7.2

Write $1$ as a fraction with a common denominator.

$| z | = \sqrt{\frac{1}{3} + \frac{3}{3}}$

Step 5.7.3

Combine the numerators over the common denominator.

$| z | = \sqrt{\frac{1 + 3}{3}}$

Step 5.7.4

Add $1$ and $3$ .

$| z | = \sqrt{\frac{4}{3}}$

Step 5.8

Rewrite $\sqrt{\frac{4}{3}}$ as $\frac{\sqrt{4}}{\sqrt{3}}$ .

$| z | = \frac{\sqrt{4}}{\sqrt{3}}$

Step 5.9

Simplify the numerator.

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Step 5.9.1

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

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

Step 5.9.2

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

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

Step 5.10

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 5.11

Combine and simplify the denominator.

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Step 5.11.1

Multiply $\frac{2}{\sqrt{3}}$ by $\frac{\sqrt{3}}{\sqrt{3}}$ .

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

Step 5.11.2

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 5.11.3

Raise $\sqrt{3}$ to the power of $1$ .

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

Step 5.11.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$| z | = \frac{2 \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 5.11.5

Add $1$ and $1$ .

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

Step 5.11.6

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 5.11.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$| z | = \frac{2 \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 5.11.6.2

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

$| z | = \frac{2 \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 5.11.6.3

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

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

Step 5.11.6.4

Cancel the common factor of $2$ .

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Step 5.11.6.4.1

Cancel the common factor.

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

Step 5.11.6.4.2

Rewrite the expression.

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

Step 5.11.6.5

Evaluate the exponent.

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

Step 6

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

$θ = \arctan (\frac{- \frac{1 \sqrt{3}}{3}}{- 1})$

Step 7

Since inverse tangent of $\frac{- \frac{1 \sqrt{3}}{3}}{- 1}$ produces an angle in the third quadrant, the value of the angle is $\frac{7 π}{6}$ .

$θ = \frac{7 π}{6}$

Step 8

Substitute the values of $θ = \frac{7 π}{6}$ and $| z | = \frac{2 \sqrt{3}}{3}$ .

$\frac{2 \sqrt{3}}{3 (\cos (\frac{7 π}{6}) + i \sin (\frac{7 π}{6}))}$