Trigonometry Examples

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

Step 1

Use the Binomial Theorem.

$3^{4} + 4 \cdot 3^{3} i + 6 \cdot 3^{2} i^{2} + 4 \cdot 3 i^{3} + i^{4}$

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Raise $3$ to the power of $4$ .

$81 + 4 \cdot 3^{3} i + 6 \cdot 3^{2} i^{2} + 4 \cdot 3 i^{3} + i^{4}$

Step 2.1.2

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

$81 + 4 \cdot 27 i + 6 \cdot 3^{2} i^{2} + 4 \cdot 3 i^{3} + i^{4}$

Step 2.1.3

Multiply $4$ by $27$ .

$81 + 108 i + 6 \cdot 3^{2} i^{2} + 4 \cdot 3 i^{3} + i^{4}$

Step 2.1.4

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

$81 + 108 i + 6 \cdot 9 i^{2} + 4 \cdot 3 i^{3} + i^{4}$

Step 2.1.5

Multiply $6$ by $9$ .

$81 + 108 i + 54 i^{2} + 4 \cdot 3 i^{3} + i^{4}$

Step 2.1.6

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

$81 + 108 i + 54 \cdot - 1 + 4 \cdot 3 i^{3} + i^{4}$

Step 2.1.7

Multiply $54$ by $- 1$ .

$81 + 108 i - 54 + 4 \cdot 3 i^{3} + i^{4}$

Step 2.1.8

Multiply $4$ by $3$ .

$81 + 108 i - 54 + 12 i^{3} + i^{4}$

Step 2.1.9

Factor out $i^{2}$ .

$81 + 108 i - 54 + 12 (i^{2} \cdot i) + i^{4}$

Step 2.1.10

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

$81 + 108 i - 54 + 12 (- 1 \cdot i) + i^{4}$

Step 2.1.11

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

$81 + 108 i - 54 + 12 (- i) + i^{4}$

Step 2.1.12

Multiply $- 1$ by $12$ .

$81 + 108 i - 54 - 12 i + i^{4}$

Step 2.1.13

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

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Step 2.1.13.1

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

$81 + 108 i - 54 - 12 i + {(i^{2})}^{2}$

Step 2.1.13.2

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

$81 + 108 i - 54 - 12 i + {(- 1)}^{2}$

Step 2.1.13.3

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

$81 + 108 i - 54 - 12 i + 1$

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $54$ from $81$ .

$27 + 108 i - 12 i + 1$

Step 2.2.2

Add $27$ and $1$ .

$28 + 108 i - 12 i$

Step 2.2.3

Subtract $12 i$ from $108 i$ .

$28 + 96 i$