Trigonometry Examples

${(- 1 + i)}^{5}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

Multiply $5$ by $1$ .

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

Step 2.1.4

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

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

Step 2.1.5

Multiply $10$ by $- 1$ .

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

Step 2.1.6

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

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

Step 2.1.7

Multiply $- 10$ by $- 1$ .

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

Step 2.1.8

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

$- 1 + 5 i + 10 + 10 \cdot 1 i^{3} + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.9

Multiply $10$ by $1$ .

$- 1 + 5 i + 10 + 10 i^{3} + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.10

Factor out $i^{2}$ .

$- 1 + 5 i + 10 + 10 (i^{2} \cdot i) + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.11

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

$- 1 + 5 i + 10 + 10 (- 1 \cdot i) + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.12

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

$- 1 + 5 i + 10 + 10 (- i) + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.13

Multiply $- 1$ by $10$ .

$- 1 + 5 i + 10 - 10 i + 5 \cdot - 1 i^{4} + i^{5}$

Step 2.1.14

Multiply $5$ by $- 1$ .

$- 1 + 5 i + 10 - 10 i - 5 i^{4} + i^{5}$

Step 2.1.15

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

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Step 2.1.15.1

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

$- 1 + 5 i + 10 - 10 i - 5 {(i^{2})}^{2} + i^{5}$

Step 2.1.15.2

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

$- 1 + 5 i + 10 - 10 i - 5 {(- 1)}^{2} + i^{5}$

Step 2.1.15.3

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

$- 1 + 5 i + 10 - 10 i - 5 \cdot 1 + i^{5}$

Step 2.1.16

Multiply $- 5$ by $1$ .

$- 1 + 5 i + 10 - 10 i - 5 + i^{5}$

Step 2.1.17

Factor out $i^{4}$ .

$- 1 + 5 i + 10 - 10 i - 5 + i^{4} i$

Step 2.1.18

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

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Step 2.1.18.1

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

$- 1 + 5 i + 10 - 10 i - 5 + {(i^{2})}^{2} i$

Step 2.1.18.2

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

$- 1 + 5 i + 10 - 10 i - 5 + {(- 1)}^{2} i$

Step 2.1.18.3

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

$- 1 + 5 i + 10 - 10 i - 5 + 1 i$

Step 2.1.19

Multiply $i$ by $1$ .

$- 1 + 5 i + 10 - 10 i - 5 + i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $- 1$ and $10$ .

$9 + 5 i - 10 i - 5 + i$

Step 2.2.2

Subtract $5$ from $9$ .

$4 + 5 i - 10 i + i$

Step 2.2.3

Subtract $10 i$ from $5 i$ .

$4 - 5 i + i$

Step 2.2.4

Add $- 5 i$ and $i$ .

$4 - 4 i$