Algebra Examples

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

One to any power is one.

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

Step 2.1.2

One to any power is one.

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

Step 2.1.3

Multiply $4$ by $1$ .

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

Step 2.1.4

One to any power is one.

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

Step 2.1.5

Multiply $6$ by $1$ .

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

Step 2.1.6

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

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

Step 2.1.7

Multiply $6$ by $- 1$ .

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

Step 2.1.8

Multiply $4$ by $1$ .

$1 + 4 i - 6 + 4 i^{3} + i^{4}$

Step 2.1.9

Factor out $i^{2}$ .

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

Step 2.1.10

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

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

Step 2.1.11

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

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

Step 2.1.12

Multiply $- 1$ by $4$ .

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

Step 2.1.13

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

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Step 2.1.13.1

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

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

Step 2.1.13.2

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

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

Step 2.1.13.3

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

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

Step 2.2

Simplify by adding terms.

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Step 2.2.1

Subtract $6$ from $1$ .

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

Step 2.2.2

Add $- 5$ and $1$ .

$- 4 + 4 i - 4 i$

Step 2.2.3

Subtract $4 i$ from $4 i$ .

$- 4 + 0$

Step 2.2.4

Add $- 4$ and $0$ .

$- 4$