Precalculus Examples

$\frac{- 5 \sqrt{3}}{2} + \frac{5}{2} i$

Step 1

Simplify each term.

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Step 1.1

Move the negative in front of the fraction.

$- \frac{(5) \sqrt{3}}{2} + \frac{5}{2} i$

Step 1.2

Combine $\frac{5}{2}$ and $i$ .

$- \frac{5 \sqrt{3}}{2} + \frac{5 i}{2}$

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 $a = - \frac{5 \sqrt{3}}{2}$ and $b = \frac{5}{2}$ .

$| z | = \sqrt{{(\frac{5}{2})}^{2} + {(- \frac{5 \sqrt{3}}{2})}^{2}}$

Step 5

Find $| z |$ .

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Step 5.1

Apply the product rule to $\frac{5}{2}$ .

$| z | = \sqrt{\frac{5^{2}}{2^{2}} + {(- \frac{5 \sqrt{3}}{2})}^{2}}$

Step 5.2

Raise $5$ to the power of $2$ .

$| z | = \sqrt{\frac{25}{2^{2}} + {(- \frac{5 \sqrt{3}}{2})}^{2}}$

Step 5.3

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

$| z | = \sqrt{\frac{25}{4} + {(- \frac{5 \sqrt{3}}{2})}^{2}}$

Step 5.4

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

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Step 5.4.1

Apply the product rule to $- \frac{5 \sqrt{3}}{2}$ .

$| z | = \sqrt{\frac{25}{4} + {(- 1)}^{2} {(\frac{5 \sqrt{3}}{2})}^{2}}$

Step 5.4.2

Apply the product rule to $\frac{5 \sqrt{3}}{2}$ .

$| z | = \sqrt{\frac{25}{4} + {(- 1)}^{2} (\frac{{(5 \sqrt{3})}^{2}}{2^{2}})}$

Step 5.4.3

Apply the product rule to $5 \sqrt{3}$ .

$| z | = \sqrt{\frac{25}{4} + {(- 1)}^{2} (\frac{5^{2} {\sqrt{3}}^{2}}{2^{2}})}$

Step 5.5

Simplify the expression.

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Step 5.5.1

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

$| z | = \sqrt{\frac{25}{4} + 1 (\frac{5^{2} {\sqrt{3}}^{2}}{2^{2}})}$

Step 5.5.2

Multiply $\frac{5^{2} {\sqrt{3}}^{2}}{2^{2}}$ by $1$ .

$| z | = \sqrt{\frac{25}{4} + \frac{5^{2} {\sqrt{3}}^{2}}{2^{2}}}$

Step 5.6

Simplify the numerator.

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Step 5.6.1

Raise $5$ to the power of $2$ .

$| z | = \sqrt{\frac{25}{4} + \frac{25 {\sqrt{3}}^{2}}{2^{2}}}$

Step 5.6.2

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

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Step 5.6.2.1

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

$| z | = \sqrt{\frac{25}{4} + \frac{25 {(3^{\frac{1}{2}})}^{2}}{2^{2}}}$

Step 5.6.2.2

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

$| z | = \sqrt{\frac{25}{4} + \frac{25 \cdot 3^{\frac{1}{2} \cdot 2}}{2^{2}}}$

Step 5.6.2.3

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

$| z | = \sqrt{\frac{25}{4} + \frac{25 \cdot 3^{\frac{2}{2}}}{2^{2}}}$

Step 5.6.2.4

Cancel the common factor of $2$ .

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Step 5.6.2.4.1

Cancel the common factor.

$| z | = \sqrt{\frac{25}{4} + \frac{25 \cdot 3^{\frac{2}{2}}}{2^{2}}}$

Step 5.6.2.4.2

Rewrite the expression.

$| z | = \sqrt{\frac{25}{4} + \frac{25 \cdot 3}{2^{2}}}$

Step 5.6.2.5

Evaluate the exponent.

$| z | = \sqrt{\frac{25}{4} + \frac{25 \cdot 3}{2^{2}}}$

Step 5.7

Simplify the expression.

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Step 5.7.1

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

$| z | = \sqrt{\frac{25}{4} + \frac{25 \cdot 3}{4}}$

Step 5.7.2

Multiply $25$ by $3$ .

$| z | = \sqrt{\frac{25}{4} + \frac{75}{4}}$

Step 5.7.3

Combine the numerators over the common denominator.

$| z | = \sqrt{\frac{25 + 75}{4}}$

Step 5.7.4

Add $25$ and $75$ .

$| z | = \sqrt{\frac{100}{4}}$

Step 5.7.5

Divide $100$ by $4$ .

$| z | = \sqrt{25}$

Step 5.7.6

Rewrite $25$ as $5^{2}$ .

$| z | = \sqrt{5^{2}}$

Step 5.8

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

$| z | = 5$

Step 6

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

$θ = \arctan (\frac{\frac{5}{2}}{- \frac{5 \sqrt{3}}{2}})$

Step 7

Since inverse tangent of $\frac{\frac{5}{2}}{- \frac{5 \sqrt{3}}{2}}$ produces an angle in the second quadrant, the value of the angle is $\frac{5 π}{6}$ .

$θ = \frac{5 π}{6}$

Step 8

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

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