Precalculus Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

One to any power is one.

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

Step 2.1.2

One to any power is one.

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

Step 2.1.3

Multiply $5$ by $1$ .

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

Step 2.1.4

Multiply $- 1$ by $5$ .

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

Step 2.1.5

One to any power is one.

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

Step 2.1.6

Multiply $10$ by $1$ .

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

Step 2.1.7

Apply the product rule to $- i$ .

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

Step 2.1.8

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

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

Step 2.1.9

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

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

Step 2.1.10

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

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

Step 2.1.11

Multiply $10$ by $- 1$ .

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

Step 2.1.12

One to any power is one.

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

Step 2.1.13

Multiply $10$ by $1$ .

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

Step 2.1.14

Apply the product rule to $- i$ .

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

Step 2.1.15

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

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

Step 2.1.16

Factor out $i^{2}$ .

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

Step 2.1.17

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

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

Step 2.1.18

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

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

Step 2.1.19

Multiply $- 1$ by $- 1$ .

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

Step 2.1.20

Multiply $i$ by $1$ .

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

Step 2.1.21

Multiply $5$ by $1$ .

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

Step 2.1.22

Apply the product rule to $- i$ .

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

Step 2.1.23

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

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

Step 2.1.24

Multiply $i^{4}$ by $1$ .

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

Step 2.1.25

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

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Step 2.1.25.1

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

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

Step 2.1.25.2

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

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

Step 2.1.25.3

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

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

Step 2.1.26

Multiply $5$ by $1$ .

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

Step 2.1.27

Apply the product rule to $- i$ .

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

Step 2.1.28

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

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

Step 2.1.29

Factor out $i^{4}$ .

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

Step 2.1.30

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

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Step 2.1.30.1

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

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

Step 2.1.30.2

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

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

Step 2.1.30.3

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

$1 - 5 i - 10 + 10 i + 5 - (1 i)$

Step 2.1.31

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

Subtract $10$ from $1$ .

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

Step 2.2.2

Add $- 9$ and $5$ .

$- 4 - 5 i + 10 i - i$

Step 2.2.3

Add $- 5 i$ and $10 i$ .

$- 4 + 5 i - i$

Step 2.2.4

Subtract $i$ from $5 i$ .

$- 4 + 4 i$