Trigonometry Examples

${(3 \sqrt{3} + 3 i)}^{5}$

Step 1

Use the Binomial Theorem.

${(3 \sqrt{3})}^{5} + 5 {(3 \sqrt{3})}^{4} (3 i) + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

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 $3 \sqrt{3}$ .

$3^{5} {\sqrt{3}}^{5} + 5 {(3 \sqrt{3})}^{4} (3 i) + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.2

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

$243 {\sqrt{3}}^{5} + 5 {(3 \sqrt{3})}^{4} (3 i) + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.3

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

$243 \sqrt{3^{5}} + 5 {(3 \sqrt{3})}^{4} (3 i) + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.4

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

$243 \sqrt{243} + 5 {(3 \sqrt{3})}^{4} (3 i) + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.5

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

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Step 2.1.5.1

Factor $81$ out of $243$ .

$243 \sqrt{81 (3)} + 5 {(3 \sqrt{3})}^{4} (3 i) + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.5.2

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

$243 \sqrt{9^{2} \cdot 3} + 5 {(3 \sqrt{3})}^{4} (3 i) + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.6

Pull terms out from under the radical.

$243 (9 \sqrt{3}) + 5 {(3 \sqrt{3})}^{4} (3 i) + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.7

Multiply $9$ by $243$ .

$2187 \sqrt{3} + 5 {(3 \sqrt{3})}^{4} (3 i) + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.8

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

$2187 \sqrt{3} + 5 (3^{4} {\sqrt{3}}^{4}) (3 i) + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.9

Multiply $3^{4}$ by $3$ by adding the exponents.

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Step 2.1.9.1

Move $3$ .

$2187 \sqrt{3} + 5 (3 \cdot 3^{4} {\sqrt{3}}^{4}) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.9.2

Multiply $3$ by $3^{4}$ .

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Step 2.1.9.2.1

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

$2187 \sqrt{3} + 5 (3^{1} \cdot 3^{4} {\sqrt{3}}^{4}) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.9.2.2

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

$2187 \sqrt{3} + 5 (3^{1 + 4} {\sqrt{3}}^{4}) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.9.3

Add $1$ and $4$ .

$2187 \sqrt{3} + 5 (3^{5} {\sqrt{3}}^{4}) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.10

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

$2187 \sqrt{3} + 5 (243 {\sqrt{3}}^{4}) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.11

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

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Step 2.1.11.1

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

$2187 \sqrt{3} + 5 (243 {(3^{\frac{1}{2}})}^{4}) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.11.2

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

$2187 \sqrt{3} + 5 (243 \cdot 3^{\frac{1}{2} \cdot 4}) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.11.3

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

$2187 \sqrt{3} + 5 (243 \cdot 3^{\frac{4}{2}}) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.11.4

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

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Step 2.1.11.4.1

Factor $2$ out of $4$ .

$2187 \sqrt{3} + 5 (243 \cdot 3^{\frac{2 \cdot 2}{2}}) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.11.4.2

Cancel the common factors.

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Step 2.1.11.4.2.1

Factor $2$ out of $2$ .

$2187 \sqrt{3} + 5 (243 \cdot 3^{\frac{2 \cdot 2}{2 (1)}}) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.11.4.2.2

Cancel the common factor.

$2187 \sqrt{3} + 5 (243 \cdot 3^{\frac{2 \cdot 2}{2 \cdot 1}}) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.11.4.2.3

Rewrite the expression.

$2187 \sqrt{3} + 5 (243 \cdot 3^{\frac{2}{1}}) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.11.4.2.4

Divide $2$ by $1$ .

$2187 \sqrt{3} + 5 (243 \cdot 3^{2}) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.12

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

$2187 \sqrt{3} + 5 (243 \cdot 9) i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.13

Multiply $5 (243 \cdot 9)$ .

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Step 2.1.13.1

Multiply $243$ by $9$ .

$2187 \sqrt{3} + 5 \cdot 2187 i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.13.2

Multiply $5$ by $2187$ .

$2187 \sqrt{3} + 10935 i + 10 {(3 \sqrt{3})}^{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.14

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

$2187 \sqrt{3} + 10935 i + 10 (3^{3} {\sqrt{3}}^{3}) {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.15

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

$2187 \sqrt{3} + 10935 i + 10 (27 {\sqrt{3}}^{3}) {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.16

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

$2187 \sqrt{3} + 10935 i + 10 (27 \sqrt{3^{3}}) {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.17

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

$2187 \sqrt{3} + 10935 i + 10 (27 \sqrt{27}) {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.18

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

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Step 2.1.18.1

Factor $9$ out of $27$ .

$2187 \sqrt{3} + 10935 i + 10 (27 \sqrt{9 (3)}) {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.18.2

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

$2187 \sqrt{3} + 10935 i + 10 (27 \sqrt{3^{2} \cdot 3}) {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.19

Pull terms out from under the radical.

$2187 \sqrt{3} + 10935 i + 10 (27 (3 \sqrt{3})) {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.20

Multiply $3$ by $27$ .

$2187 \sqrt{3} + 10935 i + 10 (81 \sqrt{3}) {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.21

Multiply $81$ by $10$ .

$2187 \sqrt{3} + 10935 i + 810 \sqrt{3} {(3 i)}^{2} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.22

Apply the product rule to $3 i$ .

$2187 \sqrt{3} + 10935 i + 810 \sqrt{3} (3^{2} i^{2}) + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.23

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

$2187 \sqrt{3} + 10935 i + 810 \sqrt{3} (9 i^{2}) + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.24

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

$2187 \sqrt{3} + 10935 i + 810 \sqrt{3} (9 \cdot - 1) + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.25

Multiply $9$ by $- 1$ .

$2187 \sqrt{3} + 10935 i + 810 \sqrt{3} \cdot - 9 + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.26

Multiply $- 9$ by $810$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 10 {(3 \sqrt{3})}^{2} {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.27

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 10 (3^{2} {\sqrt{3}}^{2}) {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.28

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 10 (9 {\sqrt{3}}^{2}) {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.29

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

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Step 2.1.29.1

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 10 (9 {(3^{\frac{1}{2}})}^{2}) {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.29.2

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 10 (9 \cdot 3^{\frac{1}{2} \cdot 2}) {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.29.3

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 10 (9 \cdot 3^{\frac{2}{2}}) {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.29.4

Cancel the common factor of $2$ .

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Step 2.1.29.4.1

Cancel the common factor.

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 10 (9 \cdot 3^{\frac{2}{2}}) {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.29.4.2

Rewrite the expression.

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 10 (9 \cdot 3^{1}) {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.29.5

Evaluate the exponent.

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 10 (9 \cdot 3) {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.30

Multiply $10 (9 \cdot 3)$ .

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Step 2.1.30.1

Multiply $9$ by $3$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 10 \cdot 27 {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.30.2

Multiply $10$ by $27$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 270 {(3 i)}^{3} + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.31

Apply the product rule to $3 i$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 270 (3^{3} i^{3}) + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.32

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 270 (27 i^{3}) + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.33

Factor out $i^{2}$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 270 (27 (i^{2} \cdot i)) + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.34

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 270 (27 (- 1 \cdot i)) + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.35

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 270 (27 (- i)) + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.36

Multiply $- 1$ by $27$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} + 270 (- 27 i) + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.37

Multiply $- 27$ by $270$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 5 (3 \sqrt{3}) {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.38

Multiply $3$ by $5$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 15 \sqrt{3} {(3 i)}^{4} + {(3 i)}^{5}$

Step 2.1.39

Apply the product rule to $3 i$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 15 \sqrt{3} (3^{4} i^{4}) + {(3 i)}^{5}$

Step 2.1.40

Raise $3$ to the power of $4$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 15 \sqrt{3} (81 i^{4}) + {(3 i)}^{5}$

Step 2.1.41

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

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Step 2.1.41.1

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 15 \sqrt{3} (81 {(i^{2})}^{2}) + {(3 i)}^{5}$

Step 2.1.41.2

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 15 \sqrt{3} (81 {(- 1)}^{2}) + {(3 i)}^{5}$

Step 2.1.41.3

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 15 \sqrt{3} (81 \cdot 1) + {(3 i)}^{5}$

Step 2.1.42

Multiply $81$ by $1$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 15 \sqrt{3} \cdot 81 + {(3 i)}^{5}$

Step 2.1.43

Multiply $81$ by $15$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 1215 \sqrt{3} + {(3 i)}^{5}$

Step 2.1.44

Apply the product rule to $3 i$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 1215 \sqrt{3} + 3^{5} i^{5}$

Step 2.1.45

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 1215 \sqrt{3} + 243 i^{5}$

Step 2.1.46

Factor out $i^{4}$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 1215 \sqrt{3} + 243 (i^{4} i)$

Step 2.1.47

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

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Step 2.1.47.1

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 1215 \sqrt{3} + 243 ({(i^{2})}^{2} i)$

Step 2.1.47.2

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 1215 \sqrt{3} + 243 ({(- 1)}^{2} i)$

Step 2.1.47.3

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

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 1215 \sqrt{3} + 243 (1 i)$

Step 2.1.48

Multiply $i$ by $1$ .

$2187 \sqrt{3} + 10935 i - 7290 \sqrt{3} - 7290 i + 1215 \sqrt{3} + 243 i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $7290 \sqrt{3}$ from $2187 \sqrt{3}$ .

$10935 i - 5103 \sqrt{3} - 7290 i + 1215 \sqrt{3} + 243 i$

Step 2.2.2

Subtract $7290 i$ from $10935 i$ .

$3645 i - 5103 \sqrt{3} + 1215 \sqrt{3} + 243 i$

Step 2.2.3

Add $3645 i$ and $243 i$ .

$3888 i - 5103 \sqrt{3} + 1215 \sqrt{3}$

Step 2.2.4

Add $- 5103 \sqrt{3}$ and $1215 \sqrt{3}$ .

$3888 i - 3888 \sqrt{3}$

Step 2.2.5

Reorder $3888 i$ and $- 3888 \sqrt{3}$ .

$- 3888 \sqrt{3} + 3888 i$