Precalculus Examples

${(- 1 + i)}^{7}$

Step 1

Use the Binomial Theorem.

${(- 1)}^{7} + 7 {(- 1)}^{6} i + 21 {(- 1)}^{5} i^{2} + 35 {(- 1)}^{4} i^{3} + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

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 $7$ .

$- 1 + 7 {(- 1)}^{6} i + 21 {(- 1)}^{5} i^{2} + 35 {(- 1)}^{4} i^{3} + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.2

Raise $- 1$ to the power of $6$ .

$- 1 + 7 \cdot 1 i + 21 {(- 1)}^{5} i^{2} + 35 {(- 1)}^{4} i^{3} + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.3

Multiply $7$ by $1$ .

$- 1 + 7 i + 21 {(- 1)}^{5} i^{2} + 35 {(- 1)}^{4} i^{3} + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.4

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

$- 1 + 7 i + 21 \cdot - 1 i^{2} + 35 {(- 1)}^{4} i^{3} + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.5

Multiply $21$ by $- 1$ .

$- 1 + 7 i - 21 i^{2} + 35 {(- 1)}^{4} i^{3} + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.6

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

$- 1 + 7 i - 21 \cdot - 1 + 35 {(- 1)}^{4} i^{3} + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.7

Multiply $- 21$ by $- 1$ .

$- 1 + 7 i + 21 + 35 {(- 1)}^{4} i^{3} + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.8

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

$- 1 + 7 i + 21 + 35 \cdot 1 i^{3} + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.9

Multiply $35$ by $1$ .

$- 1 + 7 i + 21 + 35 i^{3} + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.10

Factor out $i^{2}$ .

$- 1 + 7 i + 21 + 35 (i^{2} \cdot i) + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.11

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

$- 1 + 7 i + 21 + 35 (- 1 \cdot i) + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.12

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

$- 1 + 7 i + 21 + 35 (- i) + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.13

Multiply $- 1$ by $35$ .

$- 1 + 7 i + 21 - 35 i + 35 {(- 1)}^{3} i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.14

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

$- 1 + 7 i + 21 - 35 i + 35 \cdot - 1 i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.15

Multiply $35$ by $- 1$ .

$- 1 + 7 i + 21 - 35 i - 35 i^{4} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.16

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

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Step 2.1.16.1

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

$- 1 + 7 i + 21 - 35 i - 35 {(i^{2})}^{2} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.16.2

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

$- 1 + 7 i + 21 - 35 i - 35 {(- 1)}^{2} + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.16.3

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

$- 1 + 7 i + 21 - 35 i - 35 \cdot 1 + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.17

Multiply $- 35$ by $1$ .

$- 1 + 7 i + 21 - 35 i - 35 + 21 {(- 1)}^{2} i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.18

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 \cdot 1 i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.19

Multiply $21$ by $1$ .

$- 1 + 7 i + 21 - 35 i - 35 + 21 i^{5} + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.20

Factor out $i^{4}$ .

$- 1 + 7 i + 21 - 35 i - 35 + 21 (i^{4} i) + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.21

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

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Step 2.1.21.1

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 ({(i^{2})}^{2} i) + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.21.2

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 ({(- 1)}^{2} i) + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.21.3

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 (1 i) + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.22

Multiply $i$ by $1$ .

$- 1 + 7 i + 21 - 35 i - 35 + 21 i + 7 \cdot - 1 i^{6} + i^{7}$

Step 2.1.23

Multiply $7$ by $- 1$ .

$- 1 + 7 i + 21 - 35 i - 35 + 21 i - 7 i^{6} + i^{7}$

Step 2.1.24

Factor out $i^{4}$ .

$- 1 + 7 i + 21 - 35 i - 35 + 21 i - 7 (i^{4} i^{2}) + i^{7}$

Step 2.1.25

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

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Step 2.1.25.1

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 i - 7 ({(i^{2})}^{2} i^{2}) + i^{7}$

Step 2.1.25.2

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 i - 7 ({(- 1)}^{2} i^{2}) + i^{7}$

Step 2.1.25.3

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 i - 7 (1 i^{2}) + i^{7}$

Step 2.1.26

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 i - 7 i^{2} + i^{7}$

Step 2.1.27

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 i - 7 \cdot - 1 + i^{7}$

Step 2.1.28

Multiply $- 7$ by $- 1$ .

$- 1 + 7 i + 21 - 35 i - 35 + 21 i + 7 + i^{7}$

Step 2.1.29

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

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Step 2.1.29.1

Factor out $i^{4}$ .

$- 1 + 7 i + 21 - 35 i - 35 + 21 i + 7 + i^{4} i^{3}$

Step 2.1.29.2

Factor out $i^{2}$ .

$- 1 + 7 i + 21 - 35 i - 35 + 21 i + 7 + i^{4} (i^{2} \cdot i)$

Step 2.1.30

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

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Step 2.1.30.1

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 i + 7 + {(i^{2})}^{2} (i^{2} \cdot i)$

Step 2.1.30.2

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 i + 7 + {(- 1)}^{2} (i^{2} \cdot i)$

Step 2.1.30.3

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 i + 7 + 1 (i^{2} \cdot i)$

Step 2.1.31

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 i + 7 + i^{2} \cdot i$

Step 2.1.32

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 i + 7 - 1 \cdot i$

Step 2.1.33

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

$- 1 + 7 i + 21 - 35 i - 35 + 21 i + 7 - i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $- 1$ and $21$ .

$20 + 7 i - 35 i - 35 + 21 i + 7 - i$

Step 2.2.2

Simplify by adding and subtracting.

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Step 2.2.2.1

Subtract $35$ from $20$ .

$- 15 + 7 i - 35 i + 21 i + 7 - i$

Step 2.2.2.2

Add $- 15$ and $7$ .

$- 8 + 7 i - 35 i + 21 i - i$

Step 2.2.3

Subtract $35 i$ from $7 i$ .

$- 8 - 28 i + 21 i - i$

Step 2.2.4

Add $- 28 i$ and $21 i$ .

$- 8 - 7 i - i$

Step 2.2.5

Subtract $i$ from $- 7 i$ .

$- 8 - 8 i$