Trigonometry Examples

$3 (\cos (\frac{π}{3}) + i \sin (\frac{π}{3})) \cdot 5 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1

Simplify each term.

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Step 1.1

The exact value of $\cos (\frac{π}{3})$ is $\frac{1}{2}$ .

$3 (\frac{1}{2} + i \sin (\frac{π}{3})) \cdot 5 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1.2

The exact value of $\sin (\frac{π}{3})$ is $\frac{\sqrt{3}}{2}$ .

$3 (\frac{1}{2} + i \frac{\sqrt{3}}{2}) \cdot 5 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 1.3

Combine $i$ and $\frac{\sqrt{3}}{2}$ .

$3 (\frac{1}{2} + \frac{i \sqrt{3}}{2}) \cdot 5 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

$(3 (\frac{1}{2}) + 3 \frac{i \sqrt{3}}{2}) \cdot 5 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 2.2

Combine $3$ and $\frac{1}{2}$ .

$(\frac{3}{2} + 3 \frac{i \sqrt{3}}{2}) \cdot 5 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 2.3

Combine $3$ and $\frac{i \sqrt{3}}{2}$ .

$(\frac{3}{2} + \frac{3 (i \sqrt{3})}{2}) \cdot 5 (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 2.4

Apply the distributive property.

$(\frac{3}{2} \cdot 5 + \frac{3 i \sqrt{3}}{2} \cdot 5) (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 3

Multiply $\frac{3}{2} \cdot 5$ .

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Step 3.1

Combine $\frac{3}{2}$ and $5$ .

$(\frac{3 \cdot 5}{2} + \frac{3 i \sqrt{3}}{2} \cdot 5) (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 3.2

Multiply $3$ by $5$ .

$(\frac{15}{2} + \frac{3 i \sqrt{3}}{2} \cdot 5) (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 4

Multiply $\frac{3 i \sqrt{3}}{2} \cdot 5$ .

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Step 4.1

Combine $\frac{3 i \sqrt{3}}{2}$ and $5$ .

$(\frac{15}{2} + \frac{3 i \sqrt{3} \cdot 5}{2}) (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 4.2

Multiply $5$ by $3$ .

$(\frac{15}{2} + \frac{15 i \sqrt{3}}{2}) (\cos (\frac{π}{4}) + i \sin (\frac{π}{4}))$

Step 5

Simplify each term.

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Step 5.1

The exact value of $\cos (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$(\frac{15}{2} + \frac{15 i \sqrt{3}}{2}) (\frac{\sqrt{2}}{2} + i \sin (\frac{π}{4}))$

Step 5.2

The exact value of $\sin (\frac{π}{4})$ is $\frac{\sqrt{2}}{2}$ .

$(\frac{15}{2} + \frac{15 i \sqrt{3}}{2}) (\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2})$

Step 5.3

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

$(\frac{15}{2} + \frac{15 i \sqrt{3}}{2}) (\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})$

Step 6

Expand $(\frac{15}{2} + \frac{15 i \sqrt{3}}{2}) (\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$\frac{15}{2} (\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2}) + \frac{15 i \sqrt{3}}{2} (\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})$

Step 6.2

Apply the distributive property.

$\frac{15}{2} \cdot \frac{\sqrt{2}}{2} + \frac{15}{2} \cdot \frac{i \sqrt{2}}{2} + \frac{15 i \sqrt{3}}{2} (\frac{\sqrt{2}}{2} + \frac{i \sqrt{2}}{2})$

Step 6.3

Apply the distributive property.

$\frac{15}{2} \cdot \frac{\sqrt{2}}{2} + \frac{15}{2} \cdot \frac{i \sqrt{2}}{2} + \frac{15 i \sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{15 i \sqrt{3}}{2} \cdot \frac{i \sqrt{2}}{2}$

Step 7

Simplify terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $\frac{15}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 7.1.1.1

Multiply $\frac{15}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{15 \sqrt{2}}{2 \cdot 2} + \frac{15}{2} \cdot \frac{i \sqrt{2}}{2} + \frac{15 i \sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{15 i \sqrt{3}}{2} \cdot \frac{i \sqrt{2}}{2}$

Step 7.1.1.2

Multiply $2$ by $2$ .

$\frac{15 \sqrt{2}}{4} + \frac{15}{2} \cdot \frac{i \sqrt{2}}{2} + \frac{15 i \sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{15 i \sqrt{3}}{2} \cdot \frac{i \sqrt{2}}{2}$

Step 7.1.2

Multiply $\frac{15}{2} \cdot \frac{i \sqrt{2}}{2}$ .

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Step 7.1.2.1

Multiply $\frac{15}{2}$ by $\frac{i \sqrt{2}}{2}$ .

$\frac{15 \sqrt{2}}{4} + \frac{15 (i \sqrt{2})}{2 \cdot 2} + \frac{15 i \sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{15 i \sqrt{3}}{2} \cdot \frac{i \sqrt{2}}{2}$

Step 7.1.2.2

Multiply $2$ by $2$ .

$\frac{15 \sqrt{2}}{4} + \frac{15 (i \sqrt{2})}{4} + \frac{15 i \sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} + \frac{15 i \sqrt{3}}{2} \cdot \frac{i \sqrt{2}}{2}$

Step 7.1.3

Multiply $\frac{15 i \sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 7.1.3.1

Multiply $\frac{15 i \sqrt{3}}{2}$ by $\frac{\sqrt{2}}{2}$ .

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{3} \sqrt{2}}{2 \cdot 2} + \frac{15 i \sqrt{3}}{2} \cdot \frac{i \sqrt{2}}{2}$

Step 7.1.3.2

Combine using the product rule for radicals.

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{15 i \sqrt{3}}{2} \cdot \frac{i \sqrt{2}}{2}$

Step 7.1.3.3

Multiply $2$ by $3$ .

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{2 \cdot 2} + \frac{15 i \sqrt{3}}{2} \cdot \frac{i \sqrt{2}}{2}$

Step 7.1.3.4

Multiply $2$ by $2$ .

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4} + \frac{15 i \sqrt{3}}{2} \cdot \frac{i \sqrt{2}}{2}$

Step 7.1.4

Multiply $\frac{15 i \sqrt{3}}{2} \cdot \frac{i \sqrt{2}}{2}$ .

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Step 7.1.4.1

Multiply $\frac{15 i \sqrt{3}}{2}$ by $\frac{i \sqrt{2}}{2}$ .

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4} + \frac{15 i \sqrt{3} (i \sqrt{2})}{2 \cdot 2}$

Step 7.1.4.2

Raise $i$ to the power of $1$ .

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4} + \frac{15 (i^{1} i) \sqrt{3} \sqrt{2}}{2 \cdot 2}$

Step 7.1.4.3

Raise $i$ to the power of $1$ .

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4} + \frac{15 (i^{1} i^{1}) \sqrt{3} \sqrt{2}}{2 \cdot 2}$

Step 7.1.4.4

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

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4} + \frac{15 i^{1 + 1} \sqrt{3} \sqrt{2}}{2 \cdot 2}$

Step 7.1.4.5

Add $1$ and $1$ .

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4} + \frac{15 i^{2} \sqrt{3} \sqrt{2}}{2 \cdot 2}$

Step 7.1.4.6

Combine using the product rule for radicals.

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4} + \frac{15 i^{2} \sqrt{2 \cdot 3}}{2 \cdot 2}$

Step 7.1.4.7

Multiply $2$ by $3$ .

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4} + \frac{15 i^{2} \sqrt{6}}{2 \cdot 2}$

Step 7.1.4.8

Multiply $2$ by $2$ .

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4} + \frac{15 i^{2} \sqrt{6}}{4}$

Step 7.1.5

Simplify the numerator.

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Step 7.1.5.1

Rewrite $i^{2}$ as $- 1$ .

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4} + \frac{15 \cdot - 1 \sqrt{6}}{4}$

Step 7.1.5.2

Combine exponents.

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Step 7.1.5.2.1

Factor out negative.

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4} + \frac{- (15 \sqrt{6})}{4}$

Step 7.1.5.2.2

Multiply $15$ by $- 1$ .

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4} + \frac{- 15 \sqrt{6}}{4}$

Step 7.1.6

Move the negative in front of the fraction.

$\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4} - \frac{15 \sqrt{6}}{4}$

Step 7.2

Reorder $\frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4}$ and $- \frac{15 \sqrt{6}}{4}$ .

$- \frac{15 \sqrt{6}}{4} + \frac{15 \sqrt{2}}{4} + \frac{15 i \sqrt{2}}{4} + \frac{15 i \sqrt{6}}{4}$

Step 8

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 9

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 10

Substitute the actual values of $a = - \frac{15 \sqrt{6}}{4}$ and $b = \frac{15 \sqrt{2}}{4}$ .

$| z | = \sqrt{{(\frac{15 \sqrt{2}}{4})}^{2} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11

Find $| z |$ .

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Step 11.1

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

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Step 11.1.1

Apply the product rule to $\frac{15 \sqrt{2}}{4}$ .

$| z | = \sqrt{\frac{{(15 \sqrt{2})}^{2}}{4^{2}} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.1.2

Apply the product rule to $15 \sqrt{2}$ .

$| z | = \sqrt{\frac{15^{2} {\sqrt{2}}^{2}}{4^{2}} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.2

Simplify the numerator.

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Step 11.2.1

Raise $15$ to the power of $2$ .

$| z | = \sqrt{\frac{225 {\sqrt{2}}^{2}}{4^{2}} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.2.2

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

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Step 11.2.2.1

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

$| z | = \sqrt{\frac{225 {(2^{\frac{1}{2}})}^{2}}{4^{2}} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.2.2.2

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

$| z | = \sqrt{\frac{225 \cdot 2^{\frac{1}{2} \cdot 2}}{4^{2}} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.2.2.3

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

$| z | = \sqrt{\frac{225 \cdot 2^{\frac{2}{2}}}{4^{2}} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.2.2.4

Cancel the common factor of $2$ .

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Step 11.2.2.4.1

Cancel the common factor.

$| z | = \sqrt{\frac{225 \cdot 2^{\frac{2}{2}}}{4^{2}} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.2.2.4.2

Rewrite the expression.

$| z | = \sqrt{\frac{225 \cdot 2}{4^{2}} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.2.2.5

Evaluate the exponent.

$| z | = \sqrt{\frac{225 \cdot 2}{4^{2}} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.3

Reduce the expression by cancelling the common factors.

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Step 11.3.1

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

$| z | = \sqrt{\frac{225 \cdot 2}{16} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.3.2

Multiply $225$ by $2$ .

$| z | = \sqrt{\frac{450}{16} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.3.3

Cancel the common factor of $450$ and $16$ .

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Step 11.3.3.1

Factor $2$ out of $450$ .

$| z | = \sqrt{\frac{2 (225)}{16} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.3.3.2

Cancel the common factors.

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Step 11.3.3.2.1

Factor $2$ out of $16$ .

$| z | = \sqrt{\frac{2 \cdot 225}{2 \cdot 8} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.3.3.2.2

Cancel the common factor.

$| z | = \sqrt{\frac{2 \cdot 225}{2 \cdot 8} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.3.3.2.3

Rewrite the expression.

$| z | = \sqrt{\frac{225}{8} + {(- \frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.4

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

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Step 11.4.1

Apply the product rule to $- \frac{15 \sqrt{6}}{4}$ .

$| z | = \sqrt{\frac{225}{8} + {(- 1)}^{2} {(\frac{15 \sqrt{6}}{4})}^{2}}$

Step 11.4.2

Apply the product rule to $\frac{15 \sqrt{6}}{4}$ .

$| z | = \sqrt{\frac{225}{8} + {(- 1)}^{2} (\frac{{(15 \sqrt{6})}^{2}}{4^{2}})}$

Step 11.4.3

Apply the product rule to $15 \sqrt{6}$ .

$| z | = \sqrt{\frac{225}{8} + {(- 1)}^{2} (\frac{15^{2} {\sqrt{6}}^{2}}{4^{2}})}$

Step 11.5

Simplify the expression.

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Step 11.5.1

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

$| z | = \sqrt{\frac{225}{8} + 1 (\frac{15^{2} {\sqrt{6}}^{2}}{4^{2}})}$

Step 11.5.2

Multiply $\frac{15^{2} {\sqrt{6}}^{2}}{4^{2}}$ by $1$ .

$| z | = \sqrt{\frac{225}{8} + \frac{15^{2} {\sqrt{6}}^{2}}{4^{2}}}$

Step 11.6

Simplify the numerator.

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Step 11.6.1

Raise $15$ to the power of $2$ .

$| z | = \sqrt{\frac{225}{8} + \frac{225 {\sqrt{6}}^{2}}{4^{2}}}$

Step 11.6.2

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

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Step 11.6.2.1

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

$| z | = \sqrt{\frac{225}{8} + \frac{225 {(6^{\frac{1}{2}})}^{2}}{4^{2}}}$

Step 11.6.2.2

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

$| z | = \sqrt{\frac{225}{8} + \frac{225 \cdot 6^{\frac{1}{2} \cdot 2}}{4^{2}}}$

Step 11.6.2.3

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

$| z | = \sqrt{\frac{225}{8} + \frac{225 \cdot 6^{\frac{2}{2}}}{4^{2}}}$

Step 11.6.2.4

Cancel the common factor of $2$ .

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Step 11.6.2.4.1

Cancel the common factor.

$| z | = \sqrt{\frac{225}{8} + \frac{225 \cdot 6^{\frac{2}{2}}}{4^{2}}}$

Step 11.6.2.4.2

Rewrite the expression.

$| z | = \sqrt{\frac{225}{8} + \frac{225 \cdot 6}{4^{2}}}$

Step 11.6.2.5

Evaluate the exponent.

$| z | = \sqrt{\frac{225}{8} + \frac{225 \cdot 6}{4^{2}}}$

Step 11.7

Reduce the expression by cancelling the common factors.

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Step 11.7.1

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

$| z | = \sqrt{\frac{225}{8} + \frac{225 \cdot 6}{16}}$

Step 11.7.2

Multiply $225$ by $6$ .

$| z | = \sqrt{\frac{225}{8} + \frac{1350}{16}}$

Step 11.7.3

Cancel the common factor of $1350$ and $16$ .

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Step 11.7.3.1

Factor $2$ out of $1350$ .

$| z | = \sqrt{\frac{225}{8} + \frac{2 (675)}{16}}$

Step 11.7.3.2

Cancel the common factors.

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Step 11.7.3.2.1

Factor $2$ out of $16$ .

$| z | = \sqrt{\frac{225}{8} + \frac{2 \cdot 675}{2 \cdot 8}}$

Step 11.7.3.2.2

Cancel the common factor.

$| z | = \sqrt{\frac{225}{8} + \frac{2 \cdot 675}{2 \cdot 8}}$

Step 11.7.3.2.3

Rewrite the expression.

$| z | = \sqrt{\frac{225}{8} + \frac{675}{8}}$

Step 11.7.4

Simplify the expression.

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Step 11.7.4.1

Combine the numerators over the common denominator.

$| z | = \sqrt{\frac{225 + 675}{8}}$

Step 11.7.4.2

Add $225$ and $675$ .

$| z | = \sqrt{\frac{900}{8}}$

Step 11.7.5

Cancel the common factor of $900$ and $8$ .

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Step 11.7.5.1

Factor $4$ out of $900$ .

$| z | = \sqrt{\frac{4 (225)}{8}}$

Step 11.7.5.2

Cancel the common factors.

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Step 11.7.5.2.1

Factor $4$ out of $8$ .

$| z | = \sqrt{\frac{4 \cdot 225}{4 \cdot 2}}$

Step 11.7.5.2.2

Cancel the common factor.

$| z | = \sqrt{\frac{4 \cdot 225}{4 \cdot 2}}$

Step 11.7.5.2.3

Rewrite the expression.

$| z | = \sqrt{\frac{225}{2}}$

Step 11.8

Rewrite $\sqrt{\frac{225}{2}}$ as $\frac{\sqrt{225}}{\sqrt{2}}$ .

$| z | = \frac{\sqrt{225}}{\sqrt{2}}$

Step 11.9

Simplify the numerator.

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Step 11.9.1

Rewrite $225$ as $15^{2}$ .

$| z | = \frac{\sqrt{15^{2}}}{\sqrt{2}}$

Step 11.9.2

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

$| z | = \frac{15}{\sqrt{2}}$

Step 11.10

Multiply $\frac{15}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$| z | = \frac{15}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}$

Step 11.11

Combine and simplify the denominator.

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Step 11.11.1

Multiply $\frac{15}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$| z | = \frac{15 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 11.11.2

Raise $\sqrt{2}$ to the power of $1$ .

$| z | = \frac{15 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 11.11.3

Raise $\sqrt{2}$ to the power of $1$ .

$| z | = \frac{15 \sqrt{2}}{\sqrt{2} \sqrt{2}}$

Step 11.11.4

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

$| z | = \frac{15 \sqrt{2}}{{\sqrt{2}}^{1 + 1}}$

Step 11.11.5

Add $1$ and $1$ .

$| z | = \frac{15 \sqrt{2}}{{\sqrt{2}}^{2}}$

Step 11.11.6

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

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Step 11.11.6.1

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

$| z | = \frac{15 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}$

Step 11.11.6.2

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

$| z | = \frac{15 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}$

Step 11.11.6.3

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

$| z | = \frac{15 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 11.11.6.4

Cancel the common factor of $2$ .

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Step 11.11.6.4.1

Cancel the common factor.

$| z | = \frac{15 \sqrt{2}}{2^{\frac{2}{2}}}$

Step 11.11.6.4.2

Rewrite the expression.

$| z | = \frac{15 \sqrt{2}}{2}$

Step 11.11.6.5

Evaluate the exponent.

$| z | = \frac{15 \sqrt{2}}{2}$

Step 12

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

$θ = \arctan (\frac{\frac{15 \sqrt{2}}{4}}{- \frac{15 \sqrt{6}}{4}})$

Step 13

Since inverse tangent of $\frac{\frac{15 \sqrt{2}}{4}}{- \frac{15 \sqrt{6}}{4}}$ produces an angle in the second quadrant, the value of the angle is $\frac{5 π}{6}$ .

$θ = \frac{5 π}{6}$

Step 14

Substitute the values of $θ = \frac{5 π}{6}$ and $| z | = \frac{15 \sqrt{2}}{2}$ .

$\frac{15 \sqrt{2}}{2 (\cos (\frac{5 π}{6}) + i \sin (\frac{5 π}{6}))}$