Trigonometry Examples

$(8 (\cos (60) + i \sin (60))) (7 (\cos (165) + i \sin (165)))$

Step 1

Simplify each term.

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Step 1.1

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

$8 (\frac{1}{2} + i \sin (60)) (7 (\cos (165) + i \sin (165)))$

Step 1.2

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

$8 (\frac{1}{2} + i \frac{\sqrt{3}}{2}) (7 (\cos (165) + i \sin (165)))$

Step 1.3

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

$8 (\frac{1}{2} + \frac{i \sqrt{3}}{2}) (7 (\cos (165) + i \sin (165)))$

Step 2

Simplify terms.

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Step 2.1

Apply the distributive property.

$(8 (\frac{1}{2}) + 8 \frac{i \sqrt{3}}{2}) (7 (\cos (165) + i \sin (165)))$

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Factor $2$ out of $8$ .

$(2 (4) \frac{1}{2} + 8 \frac{i \sqrt{3}}{2}) (7 (\cos (165) + i \sin (165)))$

Step 2.2.2

Cancel the common factor.

$(2 \cdot 4 \frac{1}{2} + 8 \frac{i \sqrt{3}}{2}) (7 (\cos (165) + i \sin (165)))$

Step 2.2.3

Rewrite the expression.

$(4 + 8 \frac{i \sqrt{3}}{2}) (7 (\cos (165) + i \sin (165)))$

Step 2.3

Cancel the common factor of $2$ .

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Step 2.3.1

Factor $2$ out of $8$ .

$(4 + 2 (4) \frac{i \sqrt{3}}{2}) (7 (\cos (165) + i \sin (165)))$

Step 2.3.2

Cancel the common factor.

$(4 + 2 \cdot 4 \frac{i \sqrt{3}}{2}) (7 (\cos (165) + i \sin (165)))$

Step 2.3.3

Rewrite the expression.

$(4 + 4 (i \sqrt{3})) (7 (\cos (165) + i \sin (165)))$

Step 3

Simplify each term.

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Step 3.1

The exact value of $\cos (165)$ 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 second quadrant.

$(4 + 4 i \sqrt{3}) (7 (- \cos (15) + i \sin (165)))$

Step 3.1.2

Split $15$ into two angles where the values of the six trigonometric functions are known.

$(4 + 4 i \sqrt{3}) (7 (- \cos (45 - 30) + i \sin (165)))$

Step 3.1.3

Separate negation.

$(4 + 4 i \sqrt{3}) (7 (- \cos (45 - (30)) + i \sin (165)))$

Step 3.1.4

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

$(4 + 4 i \sqrt{3}) (7 (- (\cos (45) \cos (30) + \sin (45) \sin (30)) + i \sin (165)))$

Step 3.1.5

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

$(4 + 4 i \sqrt{3}) (7 (- (\frac{\sqrt{2}}{2} \cos (30) + \sin (45) \sin (30)) + i \sin (165)))$

Step 3.1.6

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

$(4 + 4 i \sqrt{3}) (7 (- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \sin (45) \sin (30)) + i \sin (165)))$

Step 3.1.7

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

$(4 + 4 i \sqrt{3}) (7 (- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \sin (30)) + i \sin (165)))$

Step 3.1.8

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

$(4 + 4 i \sqrt{3}) (7 (- (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + i \sin (165)))$

Step 3.1.9

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

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Step 3.1.9.1

Simplify each term.

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Step 3.1.9.1.1

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

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Step 3.1.9.1.1.1

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

$(4 + 4 i \sqrt{3}) (7 (- (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + i \sin (165)))$

Step 3.1.9.1.1.2

Combine using the product rule for radicals.

$(4 + 4 i \sqrt{3}) (7 (- (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + i \sin (165)))$

Step 3.1.9.1.1.3

Multiply $2$ by $3$ .

$(4 + 4 i \sqrt{3}) (7 (- (\frac{\sqrt{6}}{2 \cdot 2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + i \sin (165)))$

Step 3.1.9.1.1.4

Multiply $2$ by $2$ .

$(4 + 4 i \sqrt{3}) (7 (- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2}) + i \sin (165)))$

Step 3.1.9.1.2

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

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Step 3.1.9.1.2.1

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

$(4 + 4 i \sqrt{3}) (7 (- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{2 \cdot 2}) + i \sin (165)))$

Step 3.1.9.1.2.2

Multiply $2$ by $2$ .

$(4 + 4 i \sqrt{3}) (7 (- (\frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}) + i \sin (165)))$

Step 3.1.9.2

Combine the numerators over the common denominator.

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (165)))$

Step 3.2

The exact value of $\sin (165)$ is $\frac{\sqrt{6} - \sqrt{2}}{4}$ .

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Step 3.2.1

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (15)))$

Step 3.2.2

Split $15$ into two angles where the values of the six trigonometric functions are known.

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (45 - 30)))$

Step 3.2.3

Separate negation.

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i \sin (45 - (30))))$

Step 3.2.4

Apply the difference of angles identity.

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i (\sin (45) \cos (30) - \cos (45) \sin (30))))$

Step 3.2.5

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

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cos (30) - \cos (45) \sin (30))))$

Step 3.2.6

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

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \cos (45) \sin (30))))$

Step 3.2.7

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

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \sin (30))))$

Step 3.2.8

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

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))$

Step 3.2.9

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

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Step 3.2.9.1

Simplify each term.

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Step 3.2.9.1.1

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

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Step 3.2.9.1.1.1

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

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2} \sqrt{3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))$

Step 3.2.9.1.1.2

Combine using the product rule for radicals.

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{2 \cdot 3}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))$

Step 3.2.9.1.1.3

Multiply $2$ by $3$ .

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{2 \cdot 2} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))$

Step 3.2.9.1.1.4

Multiply $2$ by $2$ .

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2} \cdot \frac{1}{2})))$

Step 3.2.9.1.2

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

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Step 3.2.9.1.2.1

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

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{2 \cdot 2})))$

Step 3.2.9.1.2.2

Multiply $2$ by $2$ .

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i (\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4})))$

Step 3.2.9.2

Combine the numerators over the common denominator.

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + i \frac{\sqrt{6} - \sqrt{2}}{4}))$

Step 3.3

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

$(4 + 4 i \sqrt{3}) (7 (- \frac{\sqrt{6} + \sqrt{2}}{4} + \frac{i (\sqrt{6} - \sqrt{2})}{4}))$

Step 4

Simplify terms.

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Step 4.1

Combine the numerators over the common denominator.

$(4 + 4 i \sqrt{3}) (7 \frac{- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2}}{4})$

Step 4.2

Combine $7$ and $\frac{- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2}}{4}$ .

$(4 + 4 i \sqrt{3}) \frac{7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

Step 4.3

Apply the distributive property.

$4 \frac{7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} + 4 i \sqrt{3} \frac{7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

Step 4.4

Cancel the common factor of $4$ .

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Step 4.4.1

Cancel the common factor.

$4 \frac{7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4} + 4 i \sqrt{3} \frac{7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

Step 4.4.2

Rewrite the expression.

$7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2}) + 4 i \sqrt{3} \frac{7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

Step 4.5

Cancel the common factor of $4$ .

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Step 4.5.1

Factor $4$ out of $4 i \sqrt{3}$ .

$7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2}) + 4 (i \sqrt{3}) \frac{7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

Step 4.5.2

Cancel the common factor.

$7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2}) + 4 (i \sqrt{3}) \frac{7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2})}{4}$

Step 4.5.3

Rewrite the expression.

$7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2}) + i \sqrt{3} (7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2}))$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$7 (- \sqrt{6}) + 7 (- \sqrt{2}) + 7 (i \sqrt{6}) + 7 (- i \sqrt{2}) + i \sqrt{3} (7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2}))$

Step 5.2

Simplify.

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Step 5.2.1

Multiply $- 1$ by $7$ .

$- 7 \sqrt{6} + 7 (- \sqrt{2}) + 7 (i \sqrt{6}) + 7 (- i \sqrt{2}) + i \sqrt{3} (7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2}))$

Step 5.2.2

Multiply $- 1$ by $7$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 (i \sqrt{6}) + 7 (- i \sqrt{2}) + i \sqrt{3} (7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2}))$

Step 5.2.3

Multiply $- 1$ by $7$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 (i \sqrt{2}) + i \sqrt{3} (7 (- \sqrt{6} - \sqrt{2} + i \sqrt{6} - i \sqrt{2}))$

Step 5.3

Apply the distributive property.

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i \sqrt{3} (7 (- \sqrt{6}) + 7 (- \sqrt{2}) + 7 (i \sqrt{6}) + 7 (- i \sqrt{2}))$

Step 5.4

Simplify.

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Step 5.4.1

Multiply $- 1$ by $7$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i \sqrt{3} (- 7 \sqrt{6} + 7 (- \sqrt{2}) + 7 (i \sqrt{6}) + 7 (- i \sqrt{2}))$

Step 5.4.2

Multiply $- 1$ by $7$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i \sqrt{3} (- 7 \sqrt{6} - 7 \sqrt{2} + 7 (i \sqrt{6}) + 7 (- i \sqrt{2}))$

Step 5.4.3

Multiply $- 1$ by $7$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i \sqrt{3} (- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 (i \sqrt{2}))$

Step 5.5

Apply the distributive property.

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i \sqrt{3} (- 7 \sqrt{6}) + i \sqrt{3} (- 7 \sqrt{2}) + i \sqrt{3} (7 i \sqrt{6}) + i \sqrt{3} (- 7 i \sqrt{2})$

Step 5.6

Simplify.

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Step 5.6.1

Multiply $i \sqrt{3} (- 7 \sqrt{6})$ .

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Step 5.6.1.1

Combine using the product rule for radicals.

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{3 \cdot 6}) + i \sqrt{3} (- 7 \sqrt{2}) + i \sqrt{3} (7 i \sqrt{6}) + i \sqrt{3} (- 7 i \sqrt{2})$

Step 5.6.1.2

Multiply $3$ by $6$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i \sqrt{3} (- 7 \sqrt{2}) + i \sqrt{3} (7 i \sqrt{6}) + i \sqrt{3} (- 7 i \sqrt{2})$

Step 5.6.2

Multiply $i \sqrt{3} (- 7 \sqrt{2})$ .

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Step 5.6.2.1

Combine using the product rule for radicals.

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{3 \cdot 2}) + i \sqrt{3} (7 i \sqrt{6}) + i \sqrt{3} (- 7 i \sqrt{2})$

Step 5.6.2.2

Multiply $3$ by $2$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{6}) + i \sqrt{3} (7 i \sqrt{6}) + i \sqrt{3} (- 7 i \sqrt{2})$

Step 5.6.3

Multiply $i \sqrt{3} (7 i \sqrt{6})$ .

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Step 5.6.3.1

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

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{6}) + \sqrt{3} (7 (i^{1} i) \sqrt{6}) + i \sqrt{3} (- 7 i \sqrt{2})$

Step 5.6.3.2

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

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{6}) + \sqrt{3} (7 (i^{1} i^{1}) \sqrt{6}) + i \sqrt{3} (- 7 i \sqrt{2})$

Step 5.6.3.3

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

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{6}) + \sqrt{3} (7 i^{1 + 1} \sqrt{6}) + i \sqrt{3} (- 7 i \sqrt{2})$

Step 5.6.3.4

Add $1$ and $1$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{6}) + \sqrt{3} (7 i^{2} \sqrt{6}) + i \sqrt{3} (- 7 i \sqrt{2})$

Step 5.6.3.5

Combine using the product rule for radicals.

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{6}) + 7 i^{2} \sqrt{3 \cdot 6} + i \sqrt{3} (- 7 i \sqrt{2})$

Step 5.6.3.6

Multiply $3$ by $6$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{6}) + 7 i^{2} \sqrt{18} + i \sqrt{3} (- 7 i \sqrt{2})$

Step 5.6.4

Multiply $i \sqrt{3} (- 7 i \sqrt{2})$ .

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Step 5.6.4.1

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

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{6}) + 7 i^{2} \sqrt{18} + \sqrt{3} (- 7 (i^{1} i) \sqrt{2})$

Step 5.6.4.2

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

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{6}) + 7 i^{2} \sqrt{18} + \sqrt{3} (- 7 (i^{1} i^{1}) \sqrt{2})$

Step 5.6.4.3

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

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{6}) + 7 i^{2} \sqrt{18} + \sqrt{3} (- 7 i^{1 + 1} \sqrt{2})$

Step 5.6.4.4

Add $1$ and $1$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{6}) + 7 i^{2} \sqrt{18} + \sqrt{3} (- 7 i^{2} \sqrt{2})$

Step 5.6.4.5

Combine using the product rule for radicals.

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{6}) + 7 i^{2} \sqrt{18} - 7 i^{2} \sqrt{3 \cdot 2}$

Step 5.6.4.6

Multiply $3$ by $2$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{18}) + i (- 7 \sqrt{6}) + 7 i^{2} \sqrt{18} - 7 i^{2} \sqrt{6}$

Step 5.7

Simplify each term.

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Step 5.7.1

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

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Step 5.7.1.1

Factor $9$ out of $18$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{9 (2)}) + i (- 7 \sqrt{6}) + 7 i^{2} \sqrt{18} - 7 i^{2} \sqrt{6}$

Step 5.7.1.2

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

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 \sqrt{3^{2} \cdot 2}) + i (- 7 \sqrt{6}) + 7 i^{2} \sqrt{18} - 7 i^{2} \sqrt{6}$

Step 5.7.2

Pull terms out from under the radical.

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 7 (3 \sqrt{2})) + i (- 7 \sqrt{6}) + 7 i^{2} \sqrt{18} - 7 i^{2} \sqrt{6}$

Step 5.7.3

Multiply $3$ by $- 7$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 21 \sqrt{2}) + i (- 7 \sqrt{6}) + 7 i^{2} \sqrt{18} - 7 i^{2} \sqrt{6}$

Step 5.7.4

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

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 21 \sqrt{2}) + i (- 7 \sqrt{6}) + 7 \cdot - 1 \sqrt{18} - 7 i^{2} \sqrt{6}$

Step 5.7.5

Multiply $7$ by $- 1$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 21 \sqrt{2}) + i (- 7 \sqrt{6}) - 7 \sqrt{18} - 7 i^{2} \sqrt{6}$

Step 5.7.6

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

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Step 5.7.6.1

Factor $9$ out of $18$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 21 \sqrt{2}) + i (- 7 \sqrt{6}) - 7 \sqrt{9 (2)} - 7 i^{2} \sqrt{6}$

Step 5.7.6.2

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

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 21 \sqrt{2}) + i (- 7 \sqrt{6}) - 7 \sqrt{3^{2} \cdot 2} - 7 i^{2} \sqrt{6}$

Step 5.7.7

Pull terms out from under the radical.

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 21 \sqrt{2}) + i (- 7 \sqrt{6}) - 7 (3 \sqrt{2}) - 7 i^{2} \sqrt{6}$

Step 5.7.8

Multiply $3$ by $- 7$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 21 \sqrt{2}) + i (- 7 \sqrt{6}) - 21 \sqrt{2} - 7 i^{2} \sqrt{6}$

Step 5.7.9

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

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 21 \sqrt{2}) + i (- 7 \sqrt{6}) - 21 \sqrt{2} - 7 \cdot - 1 \sqrt{6}$

Step 5.7.10

Multiply $- 7$ by $- 1$ .

$- 7 \sqrt{6} - 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 21 \sqrt{2}) + i (- 7 \sqrt{6}) - 21 \sqrt{2} + 7 \sqrt{6}$

Step 6

Simplify by adding terms.

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Step 6.1

Add $- 7 \sqrt{6}$ and $7 \sqrt{6}$ .

$- 7 \sqrt{2} + 7 i \sqrt{6} - 7 i \sqrt{2} + i (- 21 \sqrt{2}) + i (- 7 \sqrt{6}) - 21 \sqrt{2} + 0$

Step 6.2

Subtract $21 \sqrt{2}$ from $- 7 \sqrt{2}$ .

$7 i \sqrt{6} - 7 i \sqrt{2} + i (- 21 \sqrt{2}) + i (- 7 \sqrt{6}) - 28 \sqrt{2} + 0$

Step 7

Add $7 i \sqrt{6}$ and $i (- 7 \sqrt{6})$ .

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Step 7.1

Reorder $i$ and $- 7$ .

$- 7 i \sqrt{2} + i (- 21 \sqrt{2}) + 7 i \sqrt{6} - 7 \cdot i \sqrt{6} - 28 \sqrt{2} + 0$

Step 7.2

Subtract $7 \cdot i \sqrt{6}$ from $7 i \sqrt{6}$ .

$- 7 i \sqrt{2} + i (- 21 \sqrt{2}) + 0 - 28 \sqrt{2} + 0$

Step 8

Add $- 7 i \sqrt{2}$ and $i (- 21 \sqrt{2})$ .

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Step 8.1

Reorder $i$ and $- 21$ .

$- 7 i \sqrt{2} - 21 \cdot i \sqrt{2} + 0 - 28 \sqrt{2} + 0$

Step 8.2

Subtract $21 \cdot i \sqrt{2}$ from $- 7 i \sqrt{2}$ .

$- 28 i \sqrt{2} + 0 - 28 \sqrt{2} + 0$

Step 9

Add $- 28 i \sqrt{2}$ and $0$ .

$- 28 i \sqrt{2} - 28 \sqrt{2} + 0$

Step 10

Add $- 28 i \sqrt{2}$ and $0$ .

$- 28 i \sqrt{2} - 28 \sqrt{2}$

Step 11

Reorder $- 28 i \sqrt{2}$ and $- 28 \sqrt{2}$ .

$- 28 \sqrt{2} - 28 i \sqrt{2}$