Trigonometry Examples

${(3 - 2 i)}^{8}$

Step 1

Use the Binomial Theorem.

$3^{8} + 8 \cdot 3^{7} (- 2 i) + 28 \cdot 3^{6} {(- 2 i)}^{2} + 56 \cdot 3^{5} {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

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 $8$ .

$6561 + 8 \cdot 3^{7} (- 2 i) + 28 \cdot 3^{6} {(- 2 i)}^{2} + 56 \cdot 3^{5} {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.2

Raise $3$ to the power of $7$ .

$6561 + 8 \cdot 2187 (- 2 i) + 28 \cdot 3^{6} {(- 2 i)}^{2} + 56 \cdot 3^{5} {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.3

Multiply $8$ by $2187$ .

$6561 + 17496 (- 2 i) + 28 \cdot 3^{6} {(- 2 i)}^{2} + 56 \cdot 3^{5} {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.4

Multiply $- 2$ by $17496$ .

$6561 - 34992 i + 28 \cdot 3^{6} {(- 2 i)}^{2} + 56 \cdot 3^{5} {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.5

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

$6561 - 34992 i + 28 \cdot 729 {(- 2 i)}^{2} + 56 \cdot 3^{5} {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.6

Multiply $28$ by $729$ .

$6561 - 34992 i + 20412 {(- 2 i)}^{2} + 56 \cdot 3^{5} {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.7

Apply the product rule to $- 2 i$ .

$6561 - 34992 i + 20412 ({(- 2)}^{2} i^{2}) + 56 \cdot 3^{5} {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.8

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

$6561 - 34992 i + 20412 (4 i^{2}) + 56 \cdot 3^{5} {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.9

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

$6561 - 34992 i + 20412 (4 \cdot - 1) + 56 \cdot 3^{5} {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.10

Multiply $20412 (4 \cdot - 1)$ .

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Step 2.1.10.1

Multiply $4$ by $- 1$ .

$6561 - 34992 i + 20412 \cdot - 4 + 56 \cdot 3^{5} {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.10.2

Multiply $20412$ by $- 4$ .

$6561 - 34992 i - 81648 + 56 \cdot 3^{5} {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.11

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

$6561 - 34992 i - 81648 + 56 \cdot 243 {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.12

Multiply $56$ by $243$ .

$6561 - 34992 i - 81648 + 13608 {(- 2 i)}^{3} + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.13

Apply the product rule to $- 2 i$ .

$6561 - 34992 i - 81648 + 13608 ({(- 2)}^{3} i^{3}) + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.14

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

$6561 - 34992 i - 81648 + 13608 (- 8 i^{3}) + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.15

Factor out $i^{2}$ .

$6561 - 34992 i - 81648 + 13608 (- 8 (i^{2} \cdot i)) + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.16

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

$6561 - 34992 i - 81648 + 13608 (- 8 (- 1 \cdot i)) + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.17

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

$6561 - 34992 i - 81648 + 13608 (- 8 (- i)) + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.18

Multiply $- 1$ by $- 8$ .

$6561 - 34992 i - 81648 + 13608 (8 i) + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.19

Multiply $8$ by $13608$ .

$6561 - 34992 i - 81648 + 108864 i + 70 \cdot 3^{4} {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.20

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

$6561 - 34992 i - 81648 + 108864 i + 70 \cdot 81 {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.21

Multiply $70$ by $81$ .

$6561 - 34992 i - 81648 + 108864 i + 5670 {(- 2 i)}^{4} + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.22

Apply the product rule to $- 2 i$ .

$6561 - 34992 i - 81648 + 108864 i + 5670 ({(- 2)}^{4} i^{4}) + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.23

Raise $- 2$ to the power of $4$ .

$6561 - 34992 i - 81648 + 108864 i + 5670 (16 i^{4}) + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.24

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

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Step 2.1.24.1

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

$6561 - 34992 i - 81648 + 108864 i + 5670 (16 {(i^{2})}^{2}) + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.24.2

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

$6561 - 34992 i - 81648 + 108864 i + 5670 (16 {(- 1)}^{2}) + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.24.3

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

$6561 - 34992 i - 81648 + 108864 i + 5670 (16 \cdot 1) + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.25

Multiply $5670 (16 \cdot 1)$ .

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Step 2.1.25.1

Multiply $16$ by $1$ .

$6561 - 34992 i - 81648 + 108864 i + 5670 \cdot 16 + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.25.2

Multiply $5670$ by $16$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 + 56 \cdot 3^{3} {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.26

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

$6561 - 34992 i - 81648 + 108864 i + 90720 + 56 \cdot 27 {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.27

Multiply $56$ by $27$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 + 1512 {(- 2 i)}^{5} + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.28

Apply the product rule to $- 2 i$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 + 1512 ({(- 2)}^{5} i^{5}) + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.29

Raise $- 2$ to the power of $5$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 + 1512 (- 32 i^{5}) + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.30

Factor out $i^{4}$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 + 1512 (- 32 (i^{4} i)) + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.31

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

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Step 2.1.31.1

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

$6561 - 34992 i - 81648 + 108864 i + 90720 + 1512 (- 32 ({(i^{2})}^{2} i)) + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.31.2

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

$6561 - 34992 i - 81648 + 108864 i + 90720 + 1512 (- 32 ({(- 1)}^{2} i)) + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.31.3

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

$6561 - 34992 i - 81648 + 108864 i + 90720 + 1512 (- 32 (1 i)) + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.32

Multiply $i$ by $1$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 + 1512 (- 32 i) + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.33

Multiply $- 32$ by $1512$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i + 28 \cdot 3^{2} {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.34

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i + 28 \cdot 9 {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.35

Multiply $28$ by $9$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i + 252 {(- 2 i)}^{6} + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.36

Apply the product rule to $- 2 i$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i + 252 ({(- 2)}^{6} i^{6}) + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.37

Raise $- 2$ to the power of $6$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i + 252 (64 i^{6}) + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.38

Factor out $i^{4}$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i + 252 (64 (i^{4} i^{2})) + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.39

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

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Step 2.1.39.1

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i + 252 (64 ({(i^{2})}^{2} i^{2})) + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.39.2

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i + 252 (64 ({(- 1)}^{2} i^{2})) + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.39.3

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i + 252 (64 (1 i^{2})) + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.40

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i + 252 (64 i^{2}) + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.41

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i + 252 (64 \cdot - 1) + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.42

Multiply $252 (64 \cdot - 1)$ .

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Step 2.1.42.1

Multiply $64$ by $- 1$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i + 252 \cdot - 64 + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.42.2

Multiply $252$ by $- 64$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 8 \cdot 3 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.43

Multiply $8$ by $3$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 24 {(- 2 i)}^{7} + {(- 2 i)}^{8}$

Step 2.1.44

Apply the product rule to $- 2 i$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 24 ({(- 2)}^{7} i^{7}) + {(- 2 i)}^{8}$

Step 2.1.45

Raise $- 2$ to the power of $7$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 24 (- 128 i^{7}) + {(- 2 i)}^{8}$

Step 2.1.46

Rewrite $i^{7}$ as $i^{4} (i^{2} \cdot i)$ .

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Step 2.1.46.1

Factor out $i^{4}$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 24 (- 128 (i^{4} i^{3})) + {(- 2 i)}^{8}$

Step 2.1.46.2

Factor out $i^{2}$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 24 (- 128 (i^{4} (i^{2} \cdot i))) + {(- 2 i)}^{8}$

Step 2.1.47

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

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Step 2.1.47.1

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 24 (- 128 ({(i^{2})}^{2} (i^{2} \cdot i))) + {(- 2 i)}^{8}$

Step 2.1.47.2

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 24 (- 128 ({(- 1)}^{2} (i^{2} \cdot i))) + {(- 2 i)}^{8}$

Step 2.1.47.3

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 24 (- 128 (1 (i^{2} \cdot i))) + {(- 2 i)}^{8}$

Step 2.1.48

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 24 (- 128 (i^{2} \cdot i)) + {(- 2 i)}^{8}$

Step 2.1.49

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 24 (- 128 (- 1 \cdot i)) + {(- 2 i)}^{8}$

Step 2.1.50

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 24 (- 128 (- i)) + {(- 2 i)}^{8}$

Step 2.1.51

Multiply $- 1$ by $- 128$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 24 (128 i) + {(- 2 i)}^{8}$

Step 2.1.52

Multiply $128$ by $24$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 3072 i + {(- 2 i)}^{8}$

Step 2.1.53

Apply the product rule to $- 2 i$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 3072 i + {(- 2)}^{8} i^{8}$

Step 2.1.54

Raise $- 2$ to the power of $8$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 3072 i + 256 i^{8}$

Step 2.1.55

Rewrite $i^{8}$ as ${(i^{4})}^{2}$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 3072 i + 256 {(i^{4})}^{2}$

Step 2.1.56

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

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Step 2.1.56.1

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 3072 i + 256 {({(i^{2})}^{2})}^{2}$

Step 2.1.56.2

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 3072 i + 256 {({(- 1)}^{2})}^{2}$

Step 2.1.56.3

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

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 3072 i + 256 \cdot 1^{2}$

Step 2.1.57

One to any power is one.

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 3072 i + 256 \cdot 1$

Step 2.1.58

Multiply $256$ by $1$ .

$6561 - 34992 i - 81648 + 108864 i + 90720 - 48384 i - 16128 + 3072 i + 256$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $81648$ from $6561$ .

$- 75087 - 34992 i + 108864 i + 90720 - 48384 i - 16128 + 3072 i + 256$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Add $- 75087$ and $90720$ .

$15633 - 34992 i + 108864 i - 48384 i - 16128 + 3072 i + 256$

Step 2.2.2.2

Subtract $16128$ from $15633$ .

$- 495 - 34992 i + 108864 i - 48384 i + 3072 i + 256$

Step 2.2.2.3

Add $- 495$ and $256$ .

$- 239 - 34992 i + 108864 i - 48384 i + 3072 i$

Step 2.2.3

Add $- 34992 i$ and $108864 i$ .

$- 239 + 73872 i - 48384 i + 3072 i$

Step 2.2.4

Subtract $48384 i$ from $73872 i$ .

$- 239 + 25488 i + 3072 i$

Step 2.2.5

Add $25488 i$ and $3072 i$ .

$- 239 + 28560 i$

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 $a = - 239$ and $b = 28560$ .

$| z | = \sqrt{28560^{2} + {(- 239)}^{2}}$

Step 6

Find $| z |$ .

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Step 6.1

Raise $28560$ to the power of $2$ .

$| z | = \sqrt{815673600 + {(- 239)}^{2}}$

Step 6.2

Raise $- 239$ to the power of $2$ .

$| z | = \sqrt{815673600 + 57121}$

Step 6.3

Add $815673600$ and $57121$ .

$| z | = \sqrt{815730721}$

Step 6.4

Rewrite $815730721$ as $28561^{2}$ .

$| z | = \sqrt{28561^{2}}$

Step 6.5

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

$| z | = 28561$

Step 7

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

$θ = \arctan (\frac{28560}{- 239})$

Step 8

Since inverse tangent of $\frac{28560}{- 239}$ produces an angle in the second quadrant, the value of the angle is $1.57916447$ .

$θ = 1.57916447$

Step 9

Substitute the values of $θ = 1.57916447$ and $| z | = 28561$ .

$28561 (\cos (1.57916447) + i \sin (1.57916447))$