Precalculus Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $3$ to the power of $6$ .

$729 + 6 \cdot 3^{5} (3 i \sqrt{3}) + 15 \cdot 3^{4} {(3 i \sqrt{3})}^{2} + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.2

Multiply $3^{5}$ by $3$ by adding the exponents.

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Step 2.1.2.1

Move $3$ .

$729 + 6 \cdot (3 \cdot 3^{5}) (i \sqrt{3}) + 15 \cdot 3^{4} {(3 i \sqrt{3})}^{2} + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.2.2

Multiply $3$ by $3^{5}$ .

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Step 2.1.2.2.1

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

$729 + 6 \cdot (3^{1} \cdot 3^{5}) (i \sqrt{3}) + 15 \cdot 3^{4} {(3 i \sqrt{3})}^{2} + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.2.2.2

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

$729 + 6 \cdot 3^{1 + 5} (i \sqrt{3}) + 15 \cdot 3^{4} {(3 i \sqrt{3})}^{2} + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.2.3

Add $1$ and $5$ .

$729 + 6 \cdot 3^{6} (i \sqrt{3}) + 15 \cdot 3^{4} {(3 i \sqrt{3})}^{2} + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.3

Raise $3$ to the power of $6$ .

$729 + 6 \cdot 729 (i \sqrt{3}) + 15 \cdot 3^{4} {(3 i \sqrt{3})}^{2} + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.4

Multiply $6$ by $729$ .

$729 + 4374 (i \sqrt{3}) + 15 \cdot 3^{4} {(3 i \sqrt{3})}^{2} + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.5

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

$729 + 4374 i \sqrt{3} + 15 \cdot 81 {(3 i \sqrt{3})}^{2} + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.6

Multiply $15$ by $81$ .

$729 + 4374 i \sqrt{3} + 1215 {(3 i \sqrt{3})}^{2} + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.7

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

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Step 2.1.7.1

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

$729 + 4374 i \sqrt{3} + 1215 ({(3 i)}^{2} {\sqrt{3}}^{2}) + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.7.2

Apply the product rule to $3 i$ .

$729 + 4374 i \sqrt{3} + 1215 (3^{2} i^{2} {\sqrt{3}}^{2}) + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.8

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

$729 + 4374 i \sqrt{3} + 1215 (9 i^{2} {\sqrt{3}}^{2}) + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.9

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

$729 + 4374 i \sqrt{3} + 1215 (9 \cdot - 1 {\sqrt{3}}^{2}) + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.10

Multiply $9$ by $- 1$ .

$729 + 4374 i \sqrt{3} + 1215 (- 9 {\sqrt{3}}^{2}) + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.11

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

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Step 2.1.11.1

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

$729 + 4374 i \sqrt{3} + 1215 (- 9 {(3^{\frac{1}{2}})}^{2}) + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.11.2

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

$729 + 4374 i \sqrt{3} + 1215 (- 9 \cdot 3^{\frac{1}{2} \cdot 2}) + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.11.3

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

$729 + 4374 i \sqrt{3} + 1215 (- 9 \cdot 3^{\frac{2}{2}}) + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.11.4

Cancel the common factor of $2$ .

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Step 2.1.11.4.1

Cancel the common factor.

$729 + 4374 i \sqrt{3} + 1215 (- 9 \cdot 3^{\frac{2}{2}}) + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.11.4.2

Rewrite the expression.

$729 + 4374 i \sqrt{3} + 1215 (- 9 \cdot 3^{1}) + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.11.5

Evaluate the exponent.

$729 + 4374 i \sqrt{3} + 1215 (- 9 \cdot 3) + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.12

Multiply $1215 (- 9 \cdot 3)$ .

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Step 2.1.12.1

Multiply $- 9$ by $3$ .

$729 + 4374 i \sqrt{3} + 1215 \cdot - 27 + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.12.2

Multiply $1215$ by $- 27$ .

$729 + 4374 i \sqrt{3} - 32805 + 20 \cdot 3^{3} {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.13

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

$729 + 4374 i \sqrt{3} - 32805 + 20 \cdot 27 {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.14

Multiply $20$ by $27$ .

$729 + 4374 i \sqrt{3} - 32805 + 540 {(3 i \sqrt{3})}^{3} + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.15

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

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Step 2.1.15.1

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

$729 + 4374 i \sqrt{3} - 32805 + 540 ({(3 i)}^{3} {\sqrt{3}}^{3}) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.15.2

Apply the product rule to $3 i$ .

$729 + 4374 i \sqrt{3} - 32805 + 540 (3^{3} i^{3} {\sqrt{3}}^{3}) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.16

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

$729 + 4374 i \sqrt{3} - 32805 + 540 (27 i^{3} {\sqrt{3}}^{3}) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.17

Factor out $i^{2}$ .

$729 + 4374 i \sqrt{3} - 32805 + 540 (27 (i^{2} \cdot i) {\sqrt{3}}^{3}) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.18

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

$729 + 4374 i \sqrt{3} - 32805 + 540 (27 (- 1 \cdot i) {\sqrt{3}}^{3}) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.19

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

$729 + 4374 i \sqrt{3} - 32805 + 540 (27 (- i) {\sqrt{3}}^{3}) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.20

Multiply $- 1$ by $27$ .

$729 + 4374 i \sqrt{3} - 32805 + 540 (- 27 i {\sqrt{3}}^{3}) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.21

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

$729 + 4374 i \sqrt{3} - 32805 + 540 (- 27 i \sqrt{3^{3}}) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.22

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

$729 + 4374 i \sqrt{3} - 32805 + 540 (- 27 i \sqrt{27}) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.23

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

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Step 2.1.23.1

Factor $9$ out of $27$ .

$729 + 4374 i \sqrt{3} - 32805 + 540 (- 27 i \sqrt{9 (3)}) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.23.2

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

$729 + 4374 i \sqrt{3} - 32805 + 540 (- 27 i \sqrt{3^{2} \cdot 3}) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.24

Pull terms out from under the radical.

$729 + 4374 i \sqrt{3} - 32805 + 540 (- 27 i (3 \sqrt{3})) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.25

Multiply $3$ by $- 27$ .

$729 + 4374 i \sqrt{3} - 32805 + 540 (- 81 i \sqrt{3}) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.26

Multiply $- 81$ by $540$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 (i \sqrt{3}) + 15 \cdot 3^{2} {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.27

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 15 \cdot 9 {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.28

Multiply $15$ by $9$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 {(3 i \sqrt{3})}^{4} + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.29

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

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Step 2.1.29.1

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 ({(3 i)}^{4} {\sqrt{3}}^{4}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.29.2

Apply the product rule to $3 i$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (3^{4} i^{4} {\sqrt{3}}^{4}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.30

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 i^{4} {\sqrt{3}}^{4}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.31

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

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Step 2.1.31.1

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 {(i^{2})}^{2} {\sqrt{3}}^{4}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.31.2

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 {(- 1)}^{2} {\sqrt{3}}^{4}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.31.3

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 \cdot 1 {\sqrt{3}}^{4}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.32

Multiply $81$ by $1$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 {\sqrt{3}}^{4}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.33

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

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Step 2.1.33.1

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 {(3^{\frac{1}{2}})}^{4}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.33.2

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 \cdot 3^{\frac{1}{2} \cdot 4}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.33.3

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 \cdot 3^{\frac{4}{2}}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.33.4

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

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Step 2.1.33.4.1

Factor $2$ out of $4$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 \cdot 3^{\frac{2 \cdot 2}{2}}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.33.4.2

Cancel the common factors.

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Step 2.1.33.4.2.1

Factor $2$ out of $2$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 \cdot 3^{\frac{2 \cdot 2}{2 (1)}}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.33.4.2.2

Cancel the common factor.

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 \cdot 3^{\frac{2 \cdot 2}{2 \cdot 1}}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.33.4.2.3

Rewrite the expression.

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 \cdot 3^{\frac{2}{1}}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.33.4.2.4

Divide $2$ by $1$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 \cdot 3^{2}) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.34

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 (81 \cdot 9) + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.35

Multiply $135 (81 \cdot 9)$ .

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Step 2.1.35.1

Multiply $81$ by $9$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 135 \cdot 729 + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.35.2

Multiply $135$ by $729$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 6 \cdot 3 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.36

Multiply $6$ by $3$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 {(3 i \sqrt{3})}^{5} + {(3 i \sqrt{3})}^{6}$

Step 2.1.37

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

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Step 2.1.37.1

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 ({(3 i)}^{5} {\sqrt{3}}^{5}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.37.2

Apply the product rule to $3 i$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 (3^{5} i^{5} {\sqrt{3}}^{5}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.38

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 (243 i^{5} {\sqrt{3}}^{5}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.39

Factor out $i^{4}$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 (243 (i^{4} i) {\sqrt{3}}^{5}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.40

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

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Step 2.1.40.1

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 (243 ({(i^{2})}^{2} i) {\sqrt{3}}^{5}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.40.2

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 (243 ({(- 1)}^{2} i) {\sqrt{3}}^{5}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.40.3

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 (243 (1 i) {\sqrt{3}}^{5}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.41

Multiply $i$ by $1$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 (243 i {\sqrt{3}}^{5}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.42

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 (243 i \sqrt{3^{5}}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.43

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 (243 i \sqrt{243}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.44

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

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Step 2.1.44.1

Factor $81$ out of $243$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 (243 i \sqrt{81 (3)}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.44.2

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 (243 i \sqrt{9^{2} \cdot 3}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.45

Pull terms out from under the radical.

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 (243 i (9 \sqrt{3})) + {(3 i \sqrt{3})}^{6}$

Step 2.1.46

Multiply $9$ by $243$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 18 (2187 i \sqrt{3}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.47

Multiply $2187$ by $18$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 (i \sqrt{3}) + {(3 i \sqrt{3})}^{6}$

Step 2.1.48

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

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Step 2.1.48.1

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} + {(3 i)}^{6} {\sqrt{3}}^{6}$

Step 2.1.48.2

Apply the product rule to $3 i$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} + 3^{6} i^{6} {\sqrt{3}}^{6}$

Step 2.1.49

Raise $3$ to the power of $6$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} + 729 i^{6} {\sqrt{3}}^{6}$

Step 2.1.50

Factor out $i^{4}$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} + 729 (i^{4} i^{2}) {\sqrt{3}}^{6}$

Step 2.1.51

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

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Step 2.1.51.1

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} + 729 ({(i^{2})}^{2} i^{2}) {\sqrt{3}}^{6}$

Step 2.1.51.2

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} + 729 ({(- 1)}^{2} i^{2}) {\sqrt{3}}^{6}$

Step 2.1.51.3

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} + 729 (1 i^{2}) {\sqrt{3}}^{6}$

Step 2.1.52

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} + 729 i^{2} {\sqrt{3}}^{6}$

Step 2.1.53

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} + 729 \cdot - 1 {\sqrt{3}}^{6}$

Step 2.1.54

Multiply $729$ by $- 1$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} - 729 {\sqrt{3}}^{6}$

Step 2.1.55

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

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Step 2.1.55.1

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} - 729 {(3^{\frac{1}{2}})}^{6}$

Step 2.1.55.2

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} - 729 \cdot 3^{\frac{1}{2} \cdot 6}$

Step 2.1.55.3

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} - 729 \cdot 3^{\frac{6}{2}}$

Step 2.1.55.4

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

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Step 2.1.55.4.1

Factor $2$ out of $6$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} - 729 \cdot 3^{\frac{2 \cdot 3}{2}}$

Step 2.1.55.4.2

Cancel the common factors.

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Step 2.1.55.4.2.1

Factor $2$ out of $2$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} - 729 \cdot 3^{\frac{2 \cdot 3}{2 (1)}}$

Step 2.1.55.4.2.2

Cancel the common factor.

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} - 729 \cdot 3^{\frac{2 \cdot 3}{2 \cdot 1}}$

Step 2.1.55.4.2.3

Rewrite the expression.

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} - 729 \cdot 3^{\frac{3}{1}}$

Step 2.1.55.4.2.4

Divide $3$ by $1$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} - 729 \cdot 3^{3}$

Step 2.1.56

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

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} - 729 \cdot 27$

Step 2.1.57

Multiply $- 729$ by $27$ .

$729 + 4374 i \sqrt{3} - 32805 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} - 19683$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $32805$ from $729$ .

$4374 i \sqrt{3} - 32076 - 43740 i \sqrt{3} + 98415 + 39366 i \sqrt{3} - 19683$

Step 2.2.2

Subtract $43740 i \sqrt{3}$ from $4374 i \sqrt{3}$ .

$- 39366 i \sqrt{3} - 32076 + 98415 + 39366 i \sqrt{3} - 19683$

Step 2.2.3

Add $- 39366 i \sqrt{3}$ and $39366 i \sqrt{3}$ .

$0 - 32076 + 98415 - 19683$

Step 2.2.4

Simplify by adding and subtracting.

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Step 2.2.4.1

Subtract $32076$ from $0$ .

$- 32076 + 98415 - 19683$

Step 2.2.4.2

Add $- 32076$ and $98415$ .

$66339 - 19683$

Step 2.2.4.3

Subtract $19683$ from $66339$ .

$46656$