Expand the Trigonometric Expression (1+2i)^4

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

Step 1

Use the Binomial Theorem.

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

Step 2

Simplify terms.

Simplify each term.

One to any power is one.

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

One to any power is one.

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

Multiply $4$ by $1$ .

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

Multiply $2$ by $4$ .

$1 + 8 i + 6 \cdot 1^{2} {(2 i)}^{2} + 4 \cdot 1 {(2 i)}^{3} + {(2 i)}^{4}$

One to any power is one.

$1 + 8 i + 6 \cdot 1 {(2 i)}^{2} + 4 \cdot 1 {(2 i)}^{3} + {(2 i)}^{4}$

Multiply $6$ by $1$ .

$1 + 8 i + 6 {(2 i)}^{2} + 4 \cdot 1 {(2 i)}^{3} + {(2 i)}^{4}$

Apply the product rule to $2 i$ .

$1 + 8 i + 6 (2^{2} i^{2}) + 4 \cdot 1 {(2 i)}^{3} + {(2 i)}^{4}$

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

$1 + 8 i + 6 (4 i^{2}) + 4 \cdot 1 {(2 i)}^{3} + {(2 i)}^{4}$

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

$1 + 8 i + 6 (4 \cdot - 1) + 4 \cdot 1 {(2 i)}^{3} + {(2 i)}^{4}$

Multiply $6 (4 \cdot - 1)$ .

Multiply $4$ by $- 1$ .

$1 + 8 i + 6 \cdot - 4 + 4 \cdot 1 {(2 i)}^{3} + {(2 i)}^{4}$

Multiply $6$ by $- 4$ .

$1 + 8 i - 24 + 4 \cdot 1 {(2 i)}^{3} + {(2 i)}^{4}$

Multiply $4$ by $1$ .

$1 + 8 i - 24 + 4 {(2 i)}^{3} + {(2 i)}^{4}$

Apply the product rule to $2 i$ .

$1 + 8 i - 24 + 4 (2^{3} i^{3}) + {(2 i)}^{4}$

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

$1 + 8 i - 24 + 4 (8 i^{3}) + {(2 i)}^{4}$

Factor out $i^{2}$ .

$1 + 8 i - 24 + 4 (8 (i^{2} \cdot i)) + {(2 i)}^{4}$

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

$1 + 8 i - 24 + 4 (8 (- 1 \cdot i)) + {(2 i)}^{4}$

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

$1 + 8 i - 24 + 4 (8 (- i)) + {(2 i)}^{4}$

Multiply $- 1$ by $8$ .

$1 + 8 i - 24 + 4 (- 8 i) + {(2 i)}^{4}$

Multiply $- 8$ by $4$ .

$1 + 8 i - 24 - 32 i + {(2 i)}^{4}$

Apply the product rule to $2 i$ .

$1 + 8 i - 24 - 32 i + 2^{4} i^{4}$

Raise $2$ to the power of $4$ .

$1 + 8 i - 24 - 32 i + 16 i^{4}$

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

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

$1 + 8 i - 24 - 32 i + 16 {(i^{2})}^{2}$

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

$1 + 8 i - 24 - 32 i + 16 {(- 1)}^{2}$

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

$1 + 8 i - 24 - 32 i + 16 \cdot 1$

Multiply $16$ by $1$ .

$1 + 8 i - 24 - 32 i + 16$

Simplify by adding terms.

Subtract $24$ from $1$ .

$- 23 + 8 i - 32 i + 16$

Add $- 23$ and $16$ .

$- 7 + 8 i - 32 i$

Subtract $32 i$ from $8 i$ .

$- 7 - 24 i$

Expand the Trigonometric Expression (1+2i)^4

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