Precalculus Examples

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

Step 1

Use the binomial expansion theorem to find each term. The binomial theorem states ${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

$\sum_{k = 0}^{3} \frac{3!}{(3 - k)! k!} \cdot {(3)}^{3 - k} \cdot {(- 2 i)}^{k}$

Step 2

Expand the summation.

$\frac{3!}{(3 - 0)! 0!} \cdot {(3)}^{3 - 0} \cdot {(- 2 i)}^{0} + \frac{3!}{(3 - 1)! 1!} \cdot {(3)}^{3 - 1} \cdot {(- 2 i)}^{1} + \frac{3!}{(3 - 2)! 2!} \cdot {(3)}^{3 - 2} \cdot {(- 2 i)}^{2} + \frac{3!}{(3 - 3)! 3!} \cdot {(3)}^{3 - 3} \cdot {(- 2 i)}^{3}$

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(3)}^{3} \cdot {(- 2 i)}^{0} + 3 \cdot {(3)}^{2} \cdot {(- 2 i)}^{1} + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4

Simplify the polynomial result.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply ${(3)}^{3}$ by $1$ .

${(3)}^{3} \cdot {(- 2 i)}^{0} + 3 \cdot {(3)}^{2} \cdot {(- 2 i)}^{1} + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.2

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

$27 \cdot {(- 2 i)}^{0} + 3 \cdot {(3)}^{2} \cdot {(- 2 i)}^{1} + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.3

Apply the product rule to $- 2 i$ .

$27 \cdot ({(- 2)}^{0} i^{0}) + 3 \cdot {(3)}^{2} \cdot {(- 2 i)}^{1} + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.4

Anything raised to $0$ is $1$ .

$27 \cdot (1 i^{0}) + 3 \cdot {(3)}^{2} \cdot {(- 2 i)}^{1} + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.5

Multiply $i^{0}$ by $1$ .

$27 \cdot i^{0} + 3 \cdot {(3)}^{2} \cdot {(- 2 i)}^{1} + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.6

Anything raised to $0$ is $1$ .

$27 \cdot 1 + 3 \cdot {(3)}^{2} \cdot {(- 2 i)}^{1} + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.7

Multiply $27$ by $1$ .

$27 + 3 \cdot {(3)}^{2} \cdot {(- 2 i)}^{1} + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.8

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

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Step 4.1.8.1

Multiply $3$ by ${(3)}^{2}$ .

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Step 4.1.8.1.1

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

$27 + 3^{1} \cdot {(3)}^{2} \cdot {(- 2 i)}^{1} + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.8.1.2

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

$27 + 3^{1 + 2} \cdot {(- 2 i)}^{1} + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.8.2

Add $1$ and $2$ .

$27 + 3^{3} \cdot {(- 2 i)}^{1} + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.9

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

$27 + 3^{3} \cdot (- 2 i) + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.10

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

$27 + 27 \cdot (- 2 i) + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.11

Multiply $- 2$ by $27$ .

$27 - 54 i + 3 \cdot {(3)}^{1} \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.12

Evaluate the exponent.

$27 - 54 i + 3 \cdot 3 \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.13

Multiply $3$ by $3$ .

$27 - 54 i + 9 \cdot {(- 2 i)}^{2} + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.14

Apply the product rule to $- 2 i$ .

$27 - 54 i + 9 \cdot ({(- 2)}^{2} i^{2}) + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.15

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

$27 - 54 i + 9 \cdot (4 i^{2}) + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.16

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

$27 - 54 i + 9 \cdot (4 \cdot - 1) + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.17

Multiply $9 (4 \cdot - 1)$ .

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Step 4.1.17.1

Multiply $4$ by $- 1$ .

$27 - 54 i + 9 \cdot - 4 + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.17.2

Multiply $9$ by $- 4$ .

$27 - 54 i - 36 + 1 \cdot {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.18

Multiply ${(3)}^{0}$ by $1$ .

$27 - 54 i - 36 + {(3)}^{0} \cdot {(- 2 i)}^{3}$

Step 4.1.19

Anything raised to $0$ is $1$ .

$27 - 54 i - 36 + 1 \cdot {(- 2 i)}^{3}$

Step 4.1.20

Multiply ${(- 2 i)}^{3}$ by $1$ .

$27 - 54 i - 36 + {(- 2 i)}^{3}$

Step 4.1.21

Apply the product rule to $- 2 i$ .

$27 - 54 i - 36 + {(- 2)}^{3} i^{3}$

Step 4.1.22

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

$27 - 54 i - 36 - 8 i^{3}$

Step 4.1.23

Factor out $i^{2}$ .

$27 - 54 i - 36 - 8 (i^{2} \cdot i)$

Step 4.1.24

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

$27 - 54 i - 36 - 8 (- 1 \cdot i)$

Step 4.1.25

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

$27 - 54 i - 36 - 8 (- i)$

Step 4.1.26

Multiply $- 1$ by $- 8$ .

$27 - 54 i - 36 + 8 i$

Step 4.2

Simplify by adding terms.

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Step 4.2.1

Subtract $36$ from $27$ .

$- 9 - 54 i + 8 i$

Step 4.2.2

Add $- 54 i$ and $8 i$ .

$- 9 - 46 i$