Precalculus Examples

${(- \sqrt{3} - i)}^{6}$

Step 1

Use the Binomial Theorem.

${(- \sqrt{3})}^{6} + 6 {(- \sqrt{3})}^{5} (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

${(- 1)}^{6} {\sqrt{3}}^{6} + 6 {(- \sqrt{3})}^{5} (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.2

Raise $- 1$ to the power of $6$ .

$1 {\sqrt{3}}^{6} + 6 {(- \sqrt{3})}^{5} (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.3

Multiply ${\sqrt{3}}^{6}$ by $1$ .

${\sqrt{3}}^{6} + 6 {(- \sqrt{3})}^{5} (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.4

Rewrite ${\sqrt{3}}^{6}$ as $3^{3}$ .

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Step 2.1.4.1

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

${(3^{\frac{1}{2}})}^{6} + 6 {(- \sqrt{3})}^{5} (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.4.2

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

$3^{\frac{1}{2} \cdot 6} + 6 {(- \sqrt{3})}^{5} (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.4.3

Combine $\frac{1}{2}$ and $6$ .

$3^{\frac{6}{2}} + 6 {(- \sqrt{3})}^{5} (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.4.4

Cancel the common factor of $6$ and $2$ .

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Step 2.1.4.4.1

Factor $2$ out of $6$ .

$3^{\frac{2 \cdot 3}{2}} + 6 {(- \sqrt{3})}^{5} (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.4.4.2

Cancel the common factors.

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Step 2.1.4.4.2.1

Factor $2$ out of $2$ .

$3^{\frac{2 \cdot 3}{2 (1)}} + 6 {(- \sqrt{3})}^{5} (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.4.4.2.2

Cancel the common factor.

$3^{\frac{2 \cdot 3}{2 \cdot 1}} + 6 {(- \sqrt{3})}^{5} (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.4.4.2.3

Rewrite the expression.

$3^{\frac{3}{1}} + 6 {(- \sqrt{3})}^{5} (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.4.4.2.4

Divide $3$ by $1$ .

$3^{3} + 6 {(- \sqrt{3})}^{5} (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.5

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

$27 + 6 {(- \sqrt{3})}^{5} (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.6

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

$27 + 6 ({(- 1)}^{5} {\sqrt{3}}^{5}) (- i) + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.7

Multiply ${(- 1)}^{5}$ by $- 1$ by adding the exponents.

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Step 2.1.7.1

Move $- 1$ .

$27 + 6 (- {(- 1)}^{5} {\sqrt{3}}^{5}) i + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.7.2

Multiply $- 1$ by ${(- 1)}^{5}$ .

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Step 2.1.7.2.1

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

$27 + 6 ({(- 1)}^{1} {(- 1)}^{5} {\sqrt{3}}^{5}) i + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.7.2.2

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

$27 + 6 ({(- 1)}^{1 + 5} {\sqrt{3}}^{5}) i + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.7.3

Add $1$ and $5$ .

$27 + 6 ({(- 1)}^{6} {\sqrt{3}}^{5}) i + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.8

Raise $- 1$ to the power of $6$ .

$27 + 6 (1 {\sqrt{3}}^{5}) i + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.9

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

$27 + 6 {\sqrt{3}}^{5} i + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.10

Rewrite ${\sqrt{3}}^{5}$ as $\sqrt{3^{5}}$ .

$27 + 6 \sqrt{3^{5}} i + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.11

Raise $3$ to the power of $5$ .

$27 + 6 \sqrt{243} i + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.12

Rewrite $243$ as $9^{2} \cdot 3$ .

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Step 2.1.12.1

Factor $81$ out of $243$ .

$27 + 6 \sqrt{81 (3)} i + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.12.2

Rewrite $81$ as $9^{2}$ .

$27 + 6 \sqrt{9^{2} \cdot 3} i + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.13

Pull terms out from under the radical.

$27 + 6 (9 \sqrt{3}) i + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.14

Multiply $9$ by $6$ .

$27 + 54 \sqrt{3} i + 15 {(- \sqrt{3})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.15

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

$27 + 54 \sqrt{3} i + 15 ({(- 1)}^{4} {\sqrt{3}}^{4}) {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.16

Raise $- 1$ to the power of $4$ .

$27 + 54 \sqrt{3} i + 15 (1 {\sqrt{3}}^{4}) {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.17

Multiply ${\sqrt{3}}^{4}$ by $1$ .

$27 + 54 \sqrt{3} i + 15 {\sqrt{3}}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.18

Rewrite ${\sqrt{3}}^{4}$ as $3^{2}$ .

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Step 2.1.18.1

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

$27 + 54 \sqrt{3} i + 15 {(3^{\frac{1}{2}})}^{4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.18.2

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

$27 + 54 \sqrt{3} i + 15 \cdot 3^{\frac{1}{2} \cdot 4} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.18.3

Combine $\frac{1}{2}$ and $4$ .

$27 + 54 \sqrt{3} i + 15 \cdot 3^{\frac{4}{2}} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.18.4

Cancel the common factor of $4$ and $2$ .

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Step 2.1.18.4.1

Factor $2$ out of $4$ .

$27 + 54 \sqrt{3} i + 15 \cdot 3^{\frac{2 \cdot 2}{2}} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.18.4.2

Cancel the common factors.

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Step 2.1.18.4.2.1

Factor $2$ out of $2$ .

$27 + 54 \sqrt{3} i + 15 \cdot 3^{\frac{2 \cdot 2}{2 (1)}} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.18.4.2.2

Cancel the common factor.

$27 + 54 \sqrt{3} i + 15 \cdot 3^{\frac{2 \cdot 2}{2 \cdot 1}} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.18.4.2.3

Rewrite the expression.

$27 + 54 \sqrt{3} i + 15 \cdot 3^{\frac{2}{1}} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.18.4.2.4

Divide $2$ by $1$ .

$27 + 54 \sqrt{3} i + 15 \cdot 3^{2} {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.19

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

$27 + 54 \sqrt{3} i + 15 \cdot 9 {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.20

Multiply $15$ by $9$ .

$27 + 54 \sqrt{3} i + 135 {(- i)}^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.21

Apply the product rule to $- i$ .

$27 + 54 \sqrt{3} i + 135 ({(- 1)}^{2} i^{2}) + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.22

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

$27 + 54 \sqrt{3} i + 135 (1 i^{2}) + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.23

Multiply $i^{2}$ by $1$ .

$27 + 54 \sqrt{3} i + 135 i^{2} + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.24

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

$27 + 54 \sqrt{3} i + 135 \cdot - 1 + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.25

Multiply $135$ by $- 1$ .

$27 + 54 \sqrt{3} i - 135 + 20 {(- \sqrt{3})}^{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.26

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

$27 + 54 \sqrt{3} i - 135 + 20 ({(- 1)}^{3} {\sqrt{3}}^{3}) {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.27

Raise $- 1$ to the power of $3$ .

$27 + 54 \sqrt{3} i - 135 + 20 (- {\sqrt{3}}^{3}) {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.28

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

$27 + 54 \sqrt{3} i - 135 + 20 (- \sqrt{3^{3}}) {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.29

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

$27 + 54 \sqrt{3} i - 135 + 20 (- \sqrt{27}) {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.30

Rewrite $27$ as $3^{2} \cdot 3$ .

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Step 2.1.30.1

Factor $9$ out of $27$ .

$27 + 54 \sqrt{3} i - 135 + 20 (- \sqrt{9 (3)}) {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.30.2

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

$27 + 54 \sqrt{3} i - 135 + 20 (- \sqrt{3^{2} \cdot 3}) {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.31

Pull terms out from under the radical.

$27 + 54 \sqrt{3} i - 135 + 20 (- (3 \sqrt{3})) {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.32

Multiply $3$ by $- 1$ .

$27 + 54 \sqrt{3} i - 135 + 20 (- 3 \sqrt{3}) {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.33

Multiply $- 3$ by $20$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} {(- i)}^{3} + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.34

Apply the product rule to $- i$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} ({(- 1)}^{3} i^{3}) + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.35

Raise $- 1$ to the power of $3$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} (- i^{3}) + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.36

Factor out $i^{2}$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} (- (i^{2} \cdot i)) + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.37

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} (- (- 1 \cdot i)) + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.38

Rewrite $- 1 i$ as $- i$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} (- - i) + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.39

Multiply $- 1$ by $- 1$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} (1 i) + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.40

Multiply $i$ by $1$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 15 {(- \sqrt{3})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.41

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 15 ({(- 1)}^{2} {\sqrt{3}}^{2}) {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.42

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 15 (1 {\sqrt{3}}^{2}) {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.43

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 15 {\sqrt{3}}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.44

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

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Step 2.1.44.1

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 15 {(3^{\frac{1}{2}})}^{2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.44.2

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 15 \cdot 3^{\frac{1}{2} \cdot 2} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.44.3

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 15 \cdot 3^{\frac{2}{2}} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.44.4

Cancel the common factor of $2$ .

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Step 2.1.44.4.1

Cancel the common factor.

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 15 \cdot 3^{\frac{2}{2}} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.44.4.2

Rewrite the expression.

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 15 \cdot 3^{1} {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.44.5

Evaluate the exponent.

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 15 \cdot 3 {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.45

Multiply $15$ by $3$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 {(- i)}^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.46

Apply the product rule to $- i$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 ({(- 1)}^{4} i^{4}) + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.47

Raise $- 1$ to the power of $4$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 (1 i^{4}) + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.48

Multiply $i^{4}$ by $1$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 i^{4} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.49

Rewrite $i^{4}$ as $1$ .

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Step 2.1.49.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 {(i^{2})}^{2} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.49.2

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 {(- 1)}^{2} + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.49.3

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 \cdot 1 + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.50

Multiply $45$ by $1$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 + 6 (- \sqrt{3}) {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.51

Multiply $- 1$ by $6$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 - 6 \sqrt{3} {(- i)}^{5} + {(- i)}^{6}$

Step 2.1.52

Apply the product rule to $- i$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 - 6 \sqrt{3} ({(- 1)}^{5} i^{5}) + {(- i)}^{6}$

Step 2.1.53

Raise $- 1$ to the power of $5$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 - 6 \sqrt{3} (- i^{5}) + {(- i)}^{6}$

Step 2.1.54

Factor out $i^{4}$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 - 6 \sqrt{3} (- (i^{4} i)) + {(- i)}^{6}$

Step 2.1.55

Rewrite $i^{4}$ as $1$ .

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Step 2.1.55.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 - 6 \sqrt{3} (- ({(i^{2})}^{2} i)) + {(- i)}^{6}$

Step 2.1.55.2

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 - 6 \sqrt{3} (- ({(- 1)}^{2} i)) + {(- i)}^{6}$

Step 2.1.55.3

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 - 6 \sqrt{3} (- (1 i)) + {(- i)}^{6}$

Step 2.1.56

Multiply $i$ by $1$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 - 6 \sqrt{3} (- i) + {(- i)}^{6}$

Step 2.1.57

Multiply $- 1$ by $- 6$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 + 6 \sqrt{3} i + {(- i)}^{6}$

Step 2.1.58

Apply the product rule to $- i$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 + 6 \sqrt{3} i + {(- 1)}^{6} i^{6}$

Step 2.1.59

Raise $- 1$ to the power of $6$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 + 6 \sqrt{3} i + 1 i^{6}$

Step 2.1.60

Multiply $i^{6}$ by $1$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 + 6 \sqrt{3} i + i^{6}$

Step 2.1.61

Factor out $i^{4}$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 + 6 \sqrt{3} i + i^{4} i^{2}$

Step 2.1.62

Rewrite $i^{4}$ as $1$ .

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Step 2.1.62.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 + 6 \sqrt{3} i + {(i^{2})}^{2} i^{2}$

Step 2.1.62.2

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 + 6 \sqrt{3} i + {(- 1)}^{2} i^{2}$

Step 2.1.62.3

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 + 6 \sqrt{3} i + 1 i^{2}$

Step 2.1.63

Multiply $i^{2}$ by $1$ .

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 + 6 \sqrt{3} i + i^{2}$

Step 2.1.64

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

$27 + 54 \sqrt{3} i - 135 - 60 \sqrt{3} i + 45 + 6 \sqrt{3} i - 1$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $135$ from $27$ .

$54 \sqrt{3} i - 108 - 60 \sqrt{3} i + 45 + 6 \sqrt{3} i - 1$

Step 2.2.2

Subtract $60 \sqrt{3} i$ from $54 \sqrt{3} i$ .

$- 6 \sqrt{3} i - 108 + 45 + 6 \sqrt{3} i - 1$

Step 2.2.3

Add $- 6 \sqrt{3} i$ and $6 \sqrt{3} i$ .

$0 - 108 + 45 - 1$

Step 2.2.4

Simplify by adding and subtracting.

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Step 2.2.4.1

Subtract $108$ from $0$ .

$- 108 + 45 - 1$

Step 2.2.4.2

Add $- 108$ and $45$ .

$- 63 - 1$

Step 2.2.4.3

Subtract $1$ from $- 63$ .

$- 64$

Step 3

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 4

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 5

Substitute the actual values of $a = - 64$ and $b = 0$ .

$| z | = \sqrt{0^{2} + {(- 64)}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raising $0$ to any positive power yields $0$ .

$| z | = \sqrt{0 + {(- 64)}^{2}}$

Step 6.2

Raise $- 64$ to the power of $2$ .

$| z | = \sqrt{0 + 4096}$

Step 6.3

Add $0$ and $4096$ .

$| z | = \sqrt{4096}$

Step 6.4

Rewrite $4096$ as $64^{2}$ .

$| z | = \sqrt{64^{2}}$

Step 6.5

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

$| z | = 64$

Step 7

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

$θ = \arctan (\frac{0}{- 64})$

Step 8

Since inverse tangent of $\frac{0}{- 64}$ produces an angle in the second quadrant, the value of the angle is $π$ .

$θ = π$

Step 9

Substitute the values of $θ = π$ and $| z | = 64$ .

$64 (\cos (π) + i \sin (π))$