Finite Math Examples

${(1 + i)}^{4}$

Step 1

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

$\sum_{k = 0}^{4} \frac{4!}{(4 - k)! k!} \cdot {(1)}^{4 - k} \cdot {(i)}^{k}$

Step 2

Expand the summation.

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

Step 3

Simplify the exponents for each term of the expansion.

$1 \cdot {(1)}^{4} \cdot {(i)}^{0} + 4 \cdot {(1)}^{3} \cdot {(i)}^{1} + 6 \cdot {(1)}^{2} \cdot {(i)}^{2} + 4 \cdot {(1)}^{1} \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

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 $1$ by ${(1)}^{4}$ by adding the exponents.

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Step 4.1.1.1

Multiply $1$ by ${(1)}^{4}$ .

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Step 4.1.1.1.1

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

$1^{1} \cdot {(1)}^{4} \cdot {(i)}^{0} + 4 \cdot {(1)}^{3} \cdot {(i)}^{1} + 6 \cdot {(1)}^{2} \cdot {(i)}^{2} + 4 \cdot {(1)}^{1} \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.1.1.2

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

$1^{1 + 4} \cdot {(i)}^{0} + 4 \cdot {(1)}^{3} \cdot {(i)}^{1} + 6 \cdot {(1)}^{2} \cdot {(i)}^{2} + 4 \cdot {(1)}^{1} \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.1.2

Add $1$ and $4$ .

$1^{5} \cdot {(i)}^{0} + 4 \cdot {(1)}^{3} \cdot {(i)}^{1} + 6 \cdot {(1)}^{2} \cdot {(i)}^{2} + 4 \cdot {(1)}^{1} \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.2

Simplify $1^{5} \cdot {(i)}^{0}$ .

$1^{5} + 4 \cdot {(1)}^{3} \cdot {(i)}^{1} + 6 \cdot {(1)}^{2} \cdot {(i)}^{2} + 4 \cdot {(1)}^{1} \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.3

One to any power is one.

$1 + 4 \cdot {(1)}^{3} \cdot {(i)}^{1} + 6 \cdot {(1)}^{2} \cdot {(i)}^{2} + 4 \cdot {(1)}^{1} \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.4

One to any power is one.

$1 + 4 \cdot 1 \cdot {(i)}^{1} + 6 \cdot {(1)}^{2} \cdot {(i)}^{2} + 4 \cdot {(1)}^{1} \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.5

Multiply $4$ by $1$ .

$1 + 4 \cdot {(i)}^{1} + 6 \cdot {(1)}^{2} \cdot {(i)}^{2} + 4 \cdot {(1)}^{1} \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.6

Simplify.

$1 + 4 \cdot i + 6 \cdot {(1)}^{2} \cdot {(i)}^{2} + 4 \cdot {(1)}^{1} \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.7

One to any power is one.

$1 + 4 i + 6 \cdot 1 \cdot {(i)}^{2} + 4 \cdot {(1)}^{1} \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.8

Multiply $6$ by $1$ .

$1 + 4 i + 6 \cdot {(i)}^{2} + 4 \cdot {(1)}^{1} \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.9

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

$1 + 4 i + 6 \cdot - 1 + 4 \cdot {(1)}^{1} \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.10

Multiply $6$ by $- 1$ .

$1 + 4 i - 6 + 4 \cdot {(1)}^{1} \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.11

Evaluate the exponent.

$1 + 4 i - 6 + 4 \cdot 1 \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.12

Multiply $4$ by $1$ .

$1 + 4 i - 6 + 4 \cdot {(i)}^{3} + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.13

Factor out $i^{2}$ .

$1 + 4 i - 6 + 4 \cdot (i^{2} \cdot i) + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.14

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

$1 + 4 i - 6 + 4 \cdot (- 1 \cdot i) + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.15

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

$1 + 4 i - 6 + 4 \cdot (- i) + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.16

Multiply $- 1$ by $4$ .

$1 + 4 i - 6 - 4 i + 1 \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.17

Multiply $1$ by ${(1)}^{0}$ by adding the exponents.

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Step 4.1.17.1

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

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Step 4.1.17.1.1

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

$1 + 4 i - 6 - 4 i + 1^{1} \cdot {(1)}^{0} \cdot {(i)}^{4}$

Step 4.1.17.1.2

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

$1 + 4 i - 6 - 4 i + 1^{1 + 0} \cdot {(i)}^{4}$

Step 4.1.17.2

Add $1$ and $0$ .

$1 + 4 i - 6 - 4 i + 1^{1} \cdot {(i)}^{4}$

Step 4.1.18

Simplify $1^{1} \cdot {(i)}^{4}$ .

$1 + 4 i - 6 - 4 i + {(i)}^{4}$

Step 4.1.19

Rewrite $i^{4}$ as $1$ .

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Step 4.1.19.1

Rewrite $i^{4}$ as ${(i^{2})}^{2}$ .

$1 + 4 i - 6 - 4 i + {(i^{2})}^{2}$

Step 4.1.19.2

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

$1 + 4 i - 6 - 4 i + {(- 1)}^{2}$

Step 4.1.19.3

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

$1 + 4 i - 6 - 4 i + 1$

Step 4.2

Simplify by adding terms.

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Step 4.2.1

Subtract $6$ from $1$ .

$- 5 + 4 i - 4 i + 1$

Step 4.2.2

Add $- 5$ and $1$ .

$- 4 + 4 i - 4 i$

Step 4.2.3

Subtract $4 i$ from $4 i$ .

$- 4 + 0$

Step 4.2.4

Add $- 4$ and $0$ .

$- 4$