Trigonometry Examples

${(3 - 3 i)}^{4}$

Step 1

Use the Binomial Theorem.

$3^{4} + 4 \cdot 3^{3} (- 3 i) + 6 \cdot 3^{2} {(- 3 i)}^{2} + 4 \cdot 3 {(- 3 i)}^{3} + {(- 3 i)}^{4}$

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 $4$ .

$81 + 4 \cdot 3^{3} (- 3 i) + 6 \cdot 3^{2} {(- 3 i)}^{2} + 4 \cdot 3 {(- 3 i)}^{3} + {(- 3 i)}^{4}$

Step 2.1.2

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

$81 + 4 \cdot 27 (- 3 i) + 6 \cdot 3^{2} {(- 3 i)}^{2} + 4 \cdot 3 {(- 3 i)}^{3} + {(- 3 i)}^{4}$

Step 2.1.3

Multiply $4$ by $27$ .

$81 + 108 (- 3 i) + 6 \cdot 3^{2} {(- 3 i)}^{2} + 4 \cdot 3 {(- 3 i)}^{3} + {(- 3 i)}^{4}$

Step 2.1.4

Multiply $- 3$ by $108$ .

$81 - 324 i + 6 \cdot 3^{2} {(- 3 i)}^{2} + 4 \cdot 3 {(- 3 i)}^{3} + {(- 3 i)}^{4}$

Step 2.1.5

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

$81 - 324 i + 6 \cdot 9 {(- 3 i)}^{2} + 4 \cdot 3 {(- 3 i)}^{3} + {(- 3 i)}^{4}$

Step 2.1.6

Multiply $6$ by $9$ .

$81 - 324 i + 54 {(- 3 i)}^{2} + 4 \cdot 3 {(- 3 i)}^{3} + {(- 3 i)}^{4}$

Step 2.1.7

Apply the product rule to $- 3 i$ .

$81 - 324 i + 54 ({(- 3)}^{2} i^{2}) + 4 \cdot 3 {(- 3 i)}^{3} + {(- 3 i)}^{4}$

Step 2.1.8

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

$81 - 324 i + 54 (9 i^{2}) + 4 \cdot 3 {(- 3 i)}^{3} + {(- 3 i)}^{4}$

Step 2.1.9

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

$81 - 324 i + 54 (9 \cdot - 1) + 4 \cdot 3 {(- 3 i)}^{3} + {(- 3 i)}^{4}$

Step 2.1.10

Multiply $54 (9 \cdot - 1)$ .

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Step 2.1.10.1

Multiply $9$ by $- 1$ .

$81 - 324 i + 54 \cdot - 9 + 4 \cdot 3 {(- 3 i)}^{3} + {(- 3 i)}^{4}$

Step 2.1.10.2

Multiply $54$ by $- 9$ .

$81 - 324 i - 486 + 4 \cdot 3 {(- 3 i)}^{3} + {(- 3 i)}^{4}$

Step 2.1.11

Multiply $4$ by $3$ .

$81 - 324 i - 486 + 12 {(- 3 i)}^{3} + {(- 3 i)}^{4}$

Step 2.1.12

Apply the product rule to $- 3 i$ .

$81 - 324 i - 486 + 12 ({(- 3)}^{3} i^{3}) + {(- 3 i)}^{4}$

Step 2.1.13

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

$81 - 324 i - 486 + 12 (- 27 i^{3}) + {(- 3 i)}^{4}$

Step 2.1.14

Factor out $i^{2}$ .

$81 - 324 i - 486 + 12 (- 27 (i^{2} \cdot i)) + {(- 3 i)}^{4}$

Step 2.1.15

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

$81 - 324 i - 486 + 12 (- 27 (- 1 \cdot i)) + {(- 3 i)}^{4}$

Step 2.1.16

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

$81 - 324 i - 486 + 12 (- 27 (- i)) + {(- 3 i)}^{4}$

Step 2.1.17

Multiply $- 1$ by $- 27$ .

$81 - 324 i - 486 + 12 (27 i) + {(- 3 i)}^{4}$

Step 2.1.18

Multiply $27$ by $12$ .

$81 - 324 i - 486 + 324 i + {(- 3 i)}^{4}$

Step 2.1.19

Apply the product rule to $- 3 i$ .

$81 - 324 i - 486 + 324 i + {(- 3)}^{4} i^{4}$

Step 2.1.20

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

$81 - 324 i - 486 + 324 i + 81 i^{4}$

Step 2.1.21

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

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Step 2.1.21.1

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

$81 - 324 i - 486 + 324 i + 81 {(i^{2})}^{2}$

Step 2.1.21.2

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

$81 - 324 i - 486 + 324 i + 81 {(- 1)}^{2}$

Step 2.1.21.3

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

$81 - 324 i - 486 + 324 i + 81 \cdot 1$

Step 2.1.22

Multiply $81$ by $1$ .

$81 - 324 i - 486 + 324 i + 81$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $486$ from $81$ .

$- 405 - 324 i + 324 i + 81$

Step 2.2.2

Add $- 405$ and $81$ .

$- 324 - 324 i + 324 i$

Step 2.2.3

Add $- 324 i$ and $324 i$ .

$- 324 + 0$

Step 2.2.4

Add $- 324$ and $0$ .

$- 324$