Precalculus Examples

${(- 1 - i)}^{3}$

Step 1

Use the Binomial Theorem.

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

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 $3$ .

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

Step 2.1.2

Multiply ${(- 1)}^{2}$ by $- 1$ by adding the exponents.

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Step 2.1.2.1

Move $- 1$ .

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

Step 2.1.2.2

Multiply $- 1$ by ${(- 1)}^{2}$ .

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Step 2.1.2.2.1

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

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

Step 2.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.1.2.3

Add $1$ and $2$ .

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

Step 2.1.3

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

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

Step 2.1.4

Multiply $3$ by $- 1$ .

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

Step 2.1.5

Multiply $3$ by $- 1$ .

$- 1 - 3 i - 3 {(- i)}^{2} + {(- i)}^{3}$

Step 2.1.6

Apply the product rule to $- i$ .

$- 1 - 3 i - 3 ({(- 1)}^{2} i^{2}) + {(- i)}^{3}$

Step 2.1.7

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

$- 1 - 3 i - 3 (1 i^{2}) + {(- i)}^{3}$

Step 2.1.8

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

$- 1 - 3 i - 3 i^{2} + {(- i)}^{3}$

Step 2.1.9

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

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

Step 2.1.10

Multiply $- 3$ by $- 1$ .

$- 1 - 3 i + 3 + {(- i)}^{3}$

Step 2.1.11

Apply the product rule to $- i$ .

$- 1 - 3 i + 3 + {(- 1)}^{3} i^{3}$

Step 2.1.12

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

$- 1 - 3 i + 3 - i^{3}$

Step 2.1.13

Factor out $i^{2}$ .

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

Step 2.1.14

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

$- 1 - 3 i + 3 - (- 1 \cdot i)$

Step 2.1.15

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

$- 1 - 3 i + 3 - - i$

Step 2.1.16

Multiply $- 1$ by $- 1$ .

$- 1 - 3 i + 3 + 1 i$

Step 2.1.17

Multiply $i$ by $1$ .

$- 1 - 3 i + 3 + i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $- 1$ and $3$ .

$2 - 3 i + i$

Step 2.2.2

Add $- 3 i$ and $i$ .

$2 - 2 i$