Finite Math Examples

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

Step 1

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

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

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

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

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

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

Step 4.1.2

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

$4 \cdot {(3 i)}^{0} + 2 \cdot {(2)}^{1} \cdot {(3 i)}^{1} + 1 \cdot {(2)}^{0} \cdot {(3 i)}^{2}$

Step 4.1.3

Apply the product rule to $3 i$ .

$4 \cdot (3^{0} i^{0}) + 2 \cdot {(2)}^{1} \cdot {(3 i)}^{1} + 1 \cdot {(2)}^{0} \cdot {(3 i)}^{2}$

Step 4.1.4

Anything raised to $0$ is $1$ .

$4 \cdot (1 i^{0}) + 2 \cdot {(2)}^{1} \cdot {(3 i)}^{1} + 1 \cdot {(2)}^{0} \cdot {(3 i)}^{2}$

Step 4.1.5

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

$4 \cdot i^{0} + 2 \cdot {(2)}^{1} \cdot {(3 i)}^{1} + 1 \cdot {(2)}^{0} \cdot {(3 i)}^{2}$

Step 4.1.6

Anything raised to $0$ is $1$ .

$4 \cdot 1 + 2 \cdot {(2)}^{1} \cdot {(3 i)}^{1} + 1 \cdot {(2)}^{0} \cdot {(3 i)}^{2}$

Step 4.1.7

Multiply $4$ by $1$ .

$4 + 2 \cdot {(2)}^{1} \cdot {(3 i)}^{1} + 1 \cdot {(2)}^{0} \cdot {(3 i)}^{2}$

Step 4.1.8

Evaluate the exponent.

$4 + 2 \cdot 2 \cdot {(3 i)}^{1} + 1 \cdot {(2)}^{0} \cdot {(3 i)}^{2}$

Step 4.1.9

Multiply $2$ by $2$ .

$4 + 4 \cdot {(3 i)}^{1} + 1 \cdot {(2)}^{0} \cdot {(3 i)}^{2}$

Step 4.1.10

Simplify.

$4 + 4 \cdot (3 i) + 1 \cdot {(2)}^{0} \cdot {(3 i)}^{2}$

Step 4.1.11

Multiply $3$ by $4$ .

$4 + 12 i + 1 \cdot {(2)}^{0} \cdot {(3 i)}^{2}$

Step 4.1.12

Multiply ${(2)}^{0}$ by $1$ .

$4 + 12 i + {(2)}^{0} \cdot {(3 i)}^{2}$

Step 4.1.13

Anything raised to $0$ is $1$ .

$4 + 12 i + 1 \cdot {(3 i)}^{2}$

Step 4.1.14

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

$4 + 12 i + {(3 i)}^{2}$

Step 4.1.15

Apply the product rule to $3 i$ .

$4 + 12 i + 3^{2} i^{2}$

Step 4.1.16

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

$4 + 12 i + 9 i^{2}$

Step 4.1.17

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

$4 + 12 i + 9 \cdot - 1$

Step 4.1.18

Multiply $9$ by $- 1$ .

$4 + 12 i - 9$

Step 4.2

Subtract $9$ from $4$ .

$- 5 + 12 i$