Precalculus Examples

$(2 (\cos (240) + i \sin (240))) (5 (\cos (255) + i \sin (255)))$

Step 1

Simplify each term.

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Step 1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$2 (- \cos (60) + i \sin (240)) (5 (\cos (255) + i \sin (255)))$

Step 1.2

The exact value of $\cos (60)$ is $\frac{1}{2}$ .

$2 (- \frac{1}{2} + i \sin (240)) (5 (\cos (255) + i \sin (255)))$

Step 1.3

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$2 (- \frac{1}{2} + i (- \sin (60))) (5 (\cos (255) + i \sin (255)))$

Step 1.4

The exact value of $\sin (60)$ is $\frac{\sqrt{3}}{2}$ .

$2 (- \frac{1}{2} + i (- \frac{\sqrt{3}}{2})) (5 (\cos (255) + i \sin (255)))$

Step 1.5

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

$2 (- \frac{1}{2} - \frac{i \sqrt{3}}{2}) (5 (\cos (255) + i \sin (255)))$

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

$(2 (- \frac{1}{2}) + 2 (- \frac{i \sqrt{3}}{2})) (5 (\cos (255) + i \sin (255)))$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$(2 (\frac{- 1}{2}) + 2 (- \frac{i \sqrt{3}}{2})) (5 (\cos (255) + i \sin (255)))$

Step 2.2.2

Cancel the common factor.

$(2 (\frac{- 1}{2}) + 2 (- \frac{i \sqrt{3}}{2})) (5 (\cos (255) + i \sin (255)))$

Step 2.2.3

Rewrite the expression.

$(- 1 + 2 (- \frac{i \sqrt{3}}{2})) (5 (\cos (255) + i \sin (255)))$

Step 2.3

Cancel the common factor of $2$ .

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Step 2.3.1

Move the leading negative in $- \frac{i \sqrt{3}}{2}$ into the numerator.

$(- 1 + 2 \frac{- i \sqrt{3}}{2}) (5 (\cos (255) + i \sin (255)))$

Step 2.3.2

Cancel the common factor.

$(- 1 + 2 \frac{- i \sqrt{3}}{2}) (5 (\cos (255) + i \sin (255)))$

Step 2.3.3

Rewrite the expression.

$(- 1 - i \sqrt{3}) (5 (\cos (255) + i \sin (255)))$

Step 3

Simplify each term.

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Step 3.1

The exact value of $\cos (255)$ is $- \frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 3.1.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because cosine is negative in the third quadrant.

$(- 1 - i \sqrt{3}) (5 (- \cos (75) + i \sin (255)))$

Step 3.1.2

Split $75$ into two angles where the values of the six trigonometric functions are known.

$(- 1 - i \sqrt{3}) (5 (- \cos (30 + 45) + i \sin (255)))$

Step 3.1.3

Apply the sum of angles identity $\cos (x + y) = \cos (x) \cos (y) - \sin (x) \sin (y)$ .

$(- 1 - i \sqrt{3}) (5 (- (\cos (30) \cos (45) - \sin (30) \sin (45)) + i \sin (255)))$

Step 3.1.4

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$(- 1 - i \sqrt{3}) (5 (- (\frac{\sqrt{3}}{2} \cos (45) - \sin (30) \sin (45)) + i \sin (255)))$

Step 3.1.5

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$(- 1 - i \sqrt{3}) (5 (- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \sin (30) \sin (45)) + i \sin (255)))$

Step 3.1.6

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$(- 1 - i \sqrt{3}) (5 (- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \sin (45)) + i \sin (255)))$

Step 3.1.7

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$(- 1 - i \sqrt{3}) (5 (- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + i \sin (255)))$

Step 3.1.8

Simplify $- (\frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2})$ .

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Step 3.1.8.1

Simplify each term.

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Step 3.1.8.1.1

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

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Step 3.1.8.1.1.1

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

$(- 1 - i \sqrt{3}) (5 (- (\frac{\sqrt{3} \sqrt{2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + i \sin (255)))$

Step 3.1.8.1.1.2

Combine using the product rule for radicals.

$(- 1 - i \sqrt{3}) (5 (- (\frac{\sqrt{3 \cdot 2}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + i \sin (255)))$

Step 3.1.8.1.1.3

Multiply $3$ by $2$ .

$(- 1 - i \sqrt{3}) (5 (- (\frac{\sqrt{6}}{2 \cdot 2} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + i \sin (255)))$

Step 3.1.8.1.1.4

Multiply $2$ by $2$ .

$(- 1 - i \sqrt{3}) (5 (- (\frac{\sqrt{6}}{4} - \frac{1}{2} \cdot \frac{\sqrt{2}}{2}) + i \sin (255)))$

Step 3.1.8.1.2

Multiply $- \frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 3.1.8.1.2.1

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

$(- 1 - i \sqrt{3}) (5 (- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2}) + i \sin (255)))$

Step 3.1.8.1.2.2

Multiply $2$ by $2$ .

$(- 1 - i \sqrt{3}) (5 (- (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}) + i \sin (255)))$

Step 3.1.8.2

Combine the numerators over the common denominator.

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i \sin (255)))$

Step 3.2

The exact value of $\sin (255)$ is $- \frac{\sqrt{2} + \sqrt{6}}{4}$ .

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Step 3.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because sine is negative in the third quadrant.

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- \sin (75))))$

Step 3.2.2

Split $75$ into two angles where the values of the six trigonometric functions are known.

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- \sin (30 + 45))))$

Step 3.2.3

Apply the sum of angles identity.

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- (\sin (30) \cos (45) + \cos (30) \sin (45)))))$

Step 3.2.4

The exact value of $\sin (30)$ is $\frac{1}{2}$ .

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- (\frac{1}{2} \cos (45) + \cos (30) \sin (45)))))$

Step 3.2.5

The exact value of $\cos (45)$ is $\frac{\sqrt{2}}{2}$ .

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \cos (30) \sin (45)))))$

Step 3.2.6

The exact value of $\cos (30)$ is $\frac{\sqrt{3}}{2}$ .

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \sin (45)))))$

Step 3.2.7

The exact value of $\sin (45)$ is $\frac{\sqrt{2}}{2}$ .

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}))))$

Step 3.2.8

Simplify $- (\frac{1}{2} \cdot \frac{\sqrt{2}}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2})$ .

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Step 3.2.8.1

Simplify each term.

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Step 3.2.8.1.1

Multiply $\frac{1}{2} \cdot \frac{\sqrt{2}}{2}$ .

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Step 3.2.8.1.1.1

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

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- (\frac{\sqrt{2}}{2 \cdot 2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}))))$

Step 3.2.8.1.1.2

Multiply $2$ by $2$ .

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- (\frac{\sqrt{2}}{4} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{2}}{2}))))$

Step 3.2.8.1.2

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

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Step 3.2.8.1.2.1

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

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- (\frac{\sqrt{2}}{4} + \frac{\sqrt{3} \sqrt{2}}{2 \cdot 2}))))$

Step 3.2.8.1.2.2

Combine using the product rule for radicals.

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- (\frac{\sqrt{2}}{4} + \frac{\sqrt{3 \cdot 2}}{2 \cdot 2}))))$

Step 3.2.8.1.2.3

Multiply $3$ by $2$ .

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- (\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{2 \cdot 2}))))$

Step 3.2.8.1.2.4

Multiply $2$ by $2$ .

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- (\frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4}))))$

Step 3.2.8.2

Combine the numerators over the common denominator.

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} + i (- \frac{\sqrt{2} + \sqrt{6}}{4})))$

Step 3.3

Combine $i$ and $\frac{\sqrt{2} + \sqrt{6}}{4}$ .

$(- 1 - i \sqrt{3}) (5 (- \frac{\sqrt{6} - \sqrt{2}}{4} - \frac{i (\sqrt{2} + \sqrt{6})}{4}))$

Step 4

Simplify terms.

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Step 4.1

Combine the numerators over the common denominator.

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Step 4.1.1

Apply the distributive property.

$(- 1 - i \sqrt{3}) (5 \frac{- \sqrt{6} - - \sqrt{2} - i (\sqrt{2} + \sqrt{6})}{4})$

Step 4.1.2

Multiply $- - \sqrt{2}$ .

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Step 4.1.2.1

Multiply $- 1$ by $- 1$ .

$(- 1 - i \sqrt{3}) (5 \frac{- \sqrt{6} + 1 \sqrt{2} - i (\sqrt{2} + \sqrt{6})}{4})$

Step 4.1.2.2

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

$(- 1 - i \sqrt{3}) (5 \frac{- \sqrt{6} + \sqrt{2} - i (\sqrt{2} + \sqrt{6})}{4})$

Step 4.1.3

Apply the distributive property.

$(- 1 - i \sqrt{3}) (5 \frac{- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6}}{4})$

Step 4.2

Combine $5$ and $\frac{- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6}}{4}$ .

$(- 1 - i \sqrt{3}) \frac{5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6})}{4}$

Step 4.3

Apply the distributive property.

$- 1 \frac{5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6})}{4} - i \sqrt{3} \frac{5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6})}{4}$

Step 4.4

Rewrite $- 1 \frac{5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6})}{4}$ as $- \frac{5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6})}{4}$ .

$- \frac{5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6})}{4} - i \sqrt{3} \frac{5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6})}{4}$

Step 5

Multiply $- i \sqrt{3} \frac{5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6})}{4}$ .

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Step 5.1

Combine $\frac{5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6})}{4}$ and $i$ .

$- \frac{5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6})}{4} - \frac{5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6}) i}{4} \sqrt{3}$

Step 5.2

Combine $\sqrt{3}$ and $\frac{5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6}) i}{4}$ .

$- \frac{5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6})}{4} - \frac{\sqrt{3} (5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6}) i)}{4}$

Step 6

Combine the numerators over the common denominator.

$\frac{- 5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6}) - \sqrt{3} (5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6}) i)}{4}$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$\frac{- 5 (- \sqrt{6}) - 5 \sqrt{2} - 5 (- i \sqrt{2}) - 5 (- i \sqrt{6}) - \sqrt{3} (5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6}) i)}{4}$

Step 7.2

Simplify.

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Step 7.2.1

Multiply $- 1$ by $- 5$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} - 5 (- i \sqrt{2}) - 5 (- i \sqrt{6}) - \sqrt{3} (5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6}) i)}{4}$

Step 7.2.2

Multiply $- 1$ by $- 5$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 (i \sqrt{2}) - 5 (- i \sqrt{6}) - \sqrt{3} (5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6}) i)}{4}$

Step 7.2.3

Multiply $- 1$ by $- 5$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 (i \sqrt{2}) + 5 (i \sqrt{6}) - \sqrt{3} (5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6}) i)}{4}$

Step 7.3

Remove parentheses.

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (5 (- \sqrt{6} + \sqrt{2} - i \sqrt{2} - i \sqrt{6}) i)}{4}$

Step 7.4

Apply the distributive property.

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} ((5 (- \sqrt{6}) + 5 \sqrt{2} + 5 (- i \sqrt{2}) + 5 (- i \sqrt{6})) i)}{4}$

Step 7.5

Simplify.

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Step 7.5.1

Multiply $- 1$ by $5$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} ((- 5 \sqrt{6} + 5 \sqrt{2} + 5 (- i \sqrt{2}) + 5 (- i \sqrt{6})) i)}{4}$

Step 7.5.2

Multiply $- 1$ by $5$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} ((- 5 \sqrt{6} + 5 \sqrt{2} - 5 (i \sqrt{2}) + 5 (- i \sqrt{6})) i)}{4}$

Step 7.5.3

Multiply $- 1$ by $5$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} ((- 5 \sqrt{6} + 5 \sqrt{2} - 5 (i \sqrt{2}) - 5 (i \sqrt{6})) i)}{4}$

Step 7.6

Remove parentheses.

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} ((- 5 \sqrt{6} + 5 \sqrt{2} - 5 i \sqrt{2} - 5 i \sqrt{6}) i)}{4}$

Step 7.7

Apply the distributive property.

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i + 5 \sqrt{2} i - 5 i \sqrt{2} i - 5 i \sqrt{6} i)}{4}$

Step 7.8

Simplify.

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Step 7.8.1

Multiply $- 5 i \sqrt{2} i$ .

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Step 7.8.1.1

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

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i + 5 \sqrt{2} i - 5 (i^{1} i) \sqrt{2} - 5 i \sqrt{6} i)}{4}$

Step 7.8.1.2

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

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i + 5 \sqrt{2} i - 5 (i^{1} i^{1}) \sqrt{2} - 5 i \sqrt{6} i)}{4}$

Step 7.8.1.3

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

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i + 5 \sqrt{2} i - 5 i^{1 + 1} \sqrt{2} - 5 i \sqrt{6} i)}{4}$

Step 7.8.1.4

Add $1$ and $1$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i + 5 \sqrt{2} i - 5 i^{2} \sqrt{2} - 5 i \sqrt{6} i)}{4}$

Step 7.8.2

Multiply $- 5 i \sqrt{6} i$ .

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Step 7.8.2.1

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

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i + 5 \sqrt{2} i - 5 i^{2} \sqrt{2} - 5 (i^{1} i) \sqrt{6})}{4}$

Step 7.8.2.2

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

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i + 5 \sqrt{2} i - 5 i^{2} \sqrt{2} - 5 (i^{1} i^{1}) \sqrt{6})}{4}$

Step 7.8.2.3

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

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i + 5 \sqrt{2} i - 5 i^{2} \sqrt{2} - 5 i^{1 + 1} \sqrt{6})}{4}$

Step 7.8.2.4

Add $1$ and $1$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i + 5 \sqrt{2} i - 5 i^{2} \sqrt{2} - 5 i^{2} \sqrt{6})}{4}$

Step 7.9

Simplify each term.

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Step 7.9.1

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

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i + 5 \sqrt{2} i - 5 \cdot - 1 \sqrt{2} - 5 i^{2} \sqrt{6})}{4}$

Step 7.9.2

Multiply $- 5$ by $- 1$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i + 5 \sqrt{2} i + 5 \sqrt{2} - 5 i^{2} \sqrt{6})}{4}$

Step 7.9.3

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

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i + 5 \sqrt{2} i + 5 \sqrt{2} - 5 \cdot - 1 \sqrt{6})}{4}$

Step 7.9.4

Multiply $- 5$ by $- 1$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i + 5 \sqrt{2} i + 5 \sqrt{2} + 5 \sqrt{6})}{4}$

Step 7.10

Apply the distributive property.

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} - \sqrt{3} (- 5 \sqrt{6} i) - \sqrt{3} (5 \sqrt{2} i) - \sqrt{3} (5 \sqrt{2}) - \sqrt{3} (5 \sqrt{6})}{4}$

Step 7.11

Simplify.

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Step 7.11.1

Multiply $- \sqrt{3} (- 5 \sqrt{6} i)$ .

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Step 7.11.1.1

Multiply $- 5$ by $- 1$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{3} (\sqrt{6} i) - \sqrt{3} (5 \sqrt{2} i) - \sqrt{3} (5 \sqrt{2}) - \sqrt{3} (5 \sqrt{6})}{4}$

Step 7.11.1.2

Combine using the product rule for radicals.

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{6 \cdot 3} i - \sqrt{3} (5 \sqrt{2} i) - \sqrt{3} (5 \sqrt{2}) - \sqrt{3} (5 \sqrt{6})}{4}$

Step 7.11.1.3

Multiply $6$ by $3$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{18} i - \sqrt{3} (5 \sqrt{2} i) - \sqrt{3} (5 \sqrt{2}) - \sqrt{3} (5 \sqrt{6})}{4}$

Step 7.11.2

Multiply $- \sqrt{3} (5 \sqrt{2} i)$ .

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Step 7.11.2.1

Multiply $5$ by $- 1$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{18} i - 5 \sqrt{3} (\sqrt{2} i) - \sqrt{3} (5 \sqrt{2}) - \sqrt{3} (5 \sqrt{6})}{4}$

Step 7.11.2.2

Combine using the product rule for radicals.

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{18} i - 5 \sqrt{2 \cdot 3} i - \sqrt{3} (5 \sqrt{2}) - \sqrt{3} (5 \sqrt{6})}{4}$

Step 7.11.2.3

Multiply $2$ by $3$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{18} i - 5 \sqrt{6} i - \sqrt{3} (5 \sqrt{2}) - \sqrt{3} (5 \sqrt{6})}{4}$

Step 7.11.3

Multiply $- \sqrt{3} (5 \sqrt{2})$ .

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Step 7.11.3.1

Multiply $5$ by $- 1$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{18} i - 5 \sqrt{6} i - 5 \sqrt{3} \sqrt{2} - \sqrt{3} (5 \sqrt{6})}{4}$

Step 7.11.3.2

Combine using the product rule for radicals.

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{18} i - 5 \sqrt{6} i - 5 \sqrt{2 \cdot 3} - \sqrt{3} (5 \sqrt{6})}{4}$

Step 7.11.3.3

Multiply $2$ by $3$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{18} i - 5 \sqrt{6} i - 5 \sqrt{6} - \sqrt{3} (5 \sqrt{6})}{4}$

Step 7.11.4

Multiply $- \sqrt{3} (5 \sqrt{6})$ .

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Step 7.11.4.1

Multiply $5$ by $- 1$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{18} i - 5 \sqrt{6} i - 5 \sqrt{6} - 5 \sqrt{3} \sqrt{6}}{4}$

Step 7.11.4.2

Combine using the product rule for radicals.

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{18} i - 5 \sqrt{6} i - 5 \sqrt{6} - 5 \sqrt{6 \cdot 3}}{4}$

Step 7.11.4.3

Multiply $6$ by $3$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{18} i - 5 \sqrt{6} i - 5 \sqrt{6} - 5 \sqrt{18}}{4}$

Step 7.12

Simplify each term.

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Step 7.12.1

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

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Step 7.12.1.1

Factor $9$ out of $18$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{9 (2)} i - 5 \sqrt{6} i - 5 \sqrt{6} - 5 \sqrt{18}}{4}$

Step 7.12.1.2

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

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 \sqrt{3^{2} \cdot 2} i - 5 \sqrt{6} i - 5 \sqrt{6} - 5 \sqrt{18}}{4}$

Step 7.12.2

Pull terms out from under the radical.

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 5 (3 \sqrt{2}) i - 5 \sqrt{6} i - 5 \sqrt{6} - 5 \sqrt{18}}{4}$

Step 7.12.3

Multiply $3$ by $5$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 15 \sqrt{2} i - 5 \sqrt{6} i - 5 \sqrt{6} - 5 \sqrt{18}}{4}$

Step 7.12.4

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

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Step 7.12.4.1

Factor $9$ out of $18$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 15 \sqrt{2} i - 5 \sqrt{6} i - 5 \sqrt{6} - 5 \sqrt{9 (2)}}{4}$

Step 7.12.4.2

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

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 15 \sqrt{2} i - 5 \sqrt{6} i - 5 \sqrt{6} - 5 \sqrt{3^{2} \cdot 2}}{4}$

Step 7.12.5

Pull terms out from under the radical.

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 15 \sqrt{2} i - 5 \sqrt{6} i - 5 \sqrt{6} - 5 (3 \sqrt{2})}{4}$

Step 7.12.6

Multiply $3$ by $- 5$ .

$\frac{5 \sqrt{6} - 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 15 \sqrt{2} i - 5 \sqrt{6} i - 5 \sqrt{6} - 15 \sqrt{2}}{4}$

Step 8

Simplify terms.

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Step 8.1

Subtract $5 \sqrt{6}$ from $5 \sqrt{6}$ .

$\frac{- 5 \sqrt{2} + 5 i \sqrt{2} + 5 i \sqrt{6} + 15 \sqrt{2} i - 5 \sqrt{6} i + 0 - 15 \sqrt{2}}{4}$

Step 8.2

Simplify the expression.

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Step 8.2.1

Add $- 5 \sqrt{2}$ and $0$ .

$\frac{5 i \sqrt{2} + 5 i \sqrt{6} + 15 \sqrt{2} i - 5 \sqrt{6} i - 5 \sqrt{2} - 15 \sqrt{2}}{4}$

Step 8.2.2

Reorder the factors of $15 \sqrt{2} i$ .

$\frac{5 i \sqrt{2} + 5 i \sqrt{6} + 15 i \sqrt{2} - 5 \sqrt{6} i - 5 \sqrt{2} - 15 \sqrt{2}}{4}$

Step 8.3

Add $5 i \sqrt{2}$ and $15 i \sqrt{2}$ .

$\frac{20 i \sqrt{2} + 5 i \sqrt{6} - 5 \sqrt{6} i - 5 \sqrt{2} - 15 \sqrt{2}}{4}$

Step 8.4

Reorder the factors of $- 5 \sqrt{6} i$ .

$\frac{20 i \sqrt{2} + 5 i \sqrt{6} - 5 i \sqrt{6} - 5 \sqrt{2} - 15 \sqrt{2}}{4}$

Step 8.5

Subtract $5 i \sqrt{6}$ from $5 i \sqrt{6}$ .

$\frac{20 i \sqrt{2} + 0 - 5 \sqrt{2} - 15 \sqrt{2}}{4}$

Step 8.6

Add $20 i \sqrt{2}$ and $0$ .

$\frac{20 i \sqrt{2} - 5 \sqrt{2} - 15 \sqrt{2}}{4}$

Step 8.7

Subtract $15 \sqrt{2}$ from $- 5 \sqrt{2}$ .

$\frac{20 i \sqrt{2} - 20 \sqrt{2}}{4}$

Step 8.8

Reorder $20 i \sqrt{2}$ and $- 20 \sqrt{2}$ .

$\frac{- 20 \sqrt{2} + 20 i \sqrt{2}}{4}$

Step 8.9

Factor $- 1$ out of $- 20 \sqrt{2}$ .

$\frac{- (20 \sqrt{2}) + 20 i \sqrt{2}}{4}$

Step 8.10

Factor $- 1$ out of $20 i \sqrt{2}$ .

$\frac{- (20 \sqrt{2}) - (- 20 i \sqrt{2})}{4}$

Step 8.11

Factor $- 1$ out of $- (20 \sqrt{2}) - (- 20 i \sqrt{2})$ .

$\frac{- (20 \sqrt{2} - 20 i \sqrt{2})}{4}$

Step 8.12

Simplify the expression.

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Step 8.12.1

Rewrite $- (20 \sqrt{2} - 20 i \sqrt{2})$ as $- 1 (20 \sqrt{2} - 20 i \sqrt{2})$ .

$\frac{- 1 (20 \sqrt{2} - 20 i \sqrt{2})}{4}$

Step 8.12.2

Move the negative in front of the fraction.

$- \frac{20 \sqrt{2} - 20 i \sqrt{2}}{4}$