Precalculus Examples

${(- 5 - 2 i)}^{3}$

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

$- 125 + 3 {(- 5)}^{2} (- 2 i) + 3 \cdot - 5 {(- 2 i)}^{2} + {(- 2 i)}^{3}$

Step 2.1.2

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

$- 125 + 3 \cdot 25 (- 2 i) + 3 \cdot - 5 {(- 2 i)}^{2} + {(- 2 i)}^{3}$

Step 2.1.3

Multiply $3$ by $25$ .

$- 125 + 75 (- 2 i) + 3 \cdot - 5 {(- 2 i)}^{2} + {(- 2 i)}^{3}$

Step 2.1.4

Multiply $- 2$ by $75$ .

$- 125 - 150 i + 3 \cdot - 5 {(- 2 i)}^{2} + {(- 2 i)}^{3}$

Step 2.1.5

Multiply $3$ by $- 5$ .

$- 125 - 150 i - 15 {(- 2 i)}^{2} + {(- 2 i)}^{3}$

Step 2.1.6

Apply the product rule to $- 2 i$ .

$- 125 - 150 i - 15 ({(- 2)}^{2} i^{2}) + {(- 2 i)}^{3}$

Step 2.1.7

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

$- 125 - 150 i - 15 (4 i^{2}) + {(- 2 i)}^{3}$

Step 2.1.8

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

$- 125 - 150 i - 15 (4 \cdot - 1) + {(- 2 i)}^{3}$

Step 2.1.9

Multiply $- 15 (4 \cdot - 1)$ .

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Step 2.1.9.1

Multiply $4$ by $- 1$ .

$- 125 - 150 i - 15 \cdot - 4 + {(- 2 i)}^{3}$

Step 2.1.9.2

Multiply $- 15$ by $- 4$ .

$- 125 - 150 i + 60 + {(- 2 i)}^{3}$

Step 2.1.10

Apply the product rule to $- 2 i$ .

$- 125 - 150 i + 60 + {(- 2)}^{3} i^{3}$

Step 2.1.11

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

$- 125 - 150 i + 60 - 8 i^{3}$

Step 2.1.12

Factor out $i^{2}$ .

$- 125 - 150 i + 60 - 8 (i^{2} \cdot i)$

Step 2.1.13

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

$- 125 - 150 i + 60 - 8 (- 1 \cdot i)$

Step 2.1.14

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

$- 125 - 150 i + 60 - 8 (- i)$

Step 2.1.15

Multiply $- 1$ by $- 8$ .

$- 125 - 150 i + 60 + 8 i$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Add $- 125$ and $60$ .

$- 65 - 150 i + 8 i$

Step 2.2.2

Add $- 150 i$ and $8 i$ .

$- 65 - 142 i$