Finite Math Examples

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

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

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})

${(a + b)}^{n} = \sum_{k = 0}^{n} n C k \cdot (a^{n - k} b^{k})$ .

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

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

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}

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

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}

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

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}

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

1

$1$ .

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

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

Step 4.1.2

Raise

2

$2$ to the power of

2

$2$ .

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

$3 i$ .

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

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

Step 4.1.4

Anything raised to

0

$0$ is

1

$1$ .

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

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

Step 4.1.5

Multiply

i^{0}

$i^{0}$ by

1

$1$ .

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

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

Step 4.1.6

Anything raised to

0

$0$ is

1

$1$ .

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

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

Step 4.1.7

Multiply

4

$4$ by

1

$1$ .

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

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

Step 4.1.8

Evaluate the exponent.

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

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

Step 4.1.9

Multiply

2

$2$ by

2

$2$ .

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

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

Step 4.1.11

Multiply

3

$3$ by

4

$4$ .

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

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

Step 4.1.12

Multiply

{(2)}^{0}

${(2)}^{0}$ by

1

$1$ .

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

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

Step 4.1.13

Anything raised to

0

$0$ is

1

$1$ .

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

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

Step 4.1.14

Multiply

{(3 i)}^{2}

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

1

$1$ .

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

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

Step 4.1.15

Apply the product rule to

3 i

$3 i$ .

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

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

Step 4.1.16

Raise

3

$3$ to the power of

2

$2$ .

4 + 12 i + 9 i^{2}

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

Step 4.1.17

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

4 + 12 i + 9 \cdot - 1

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

Step 4.1.18

Multiply

9

$9$ by

- 1

$- 1$ .

4 + 12 i - 9

$4 + 12 i - 9$

4 + 12 i - 9

$4 + 12 i - 9$

Step 4.2

Subtract

9

$9$ from

4

$4$ .

- 5 + 12 i

$- 5 + 12 i$

- 5 + 12 i

$- 5 + 12 i$