Simplify (1+zi)(1+2i)-(1+zi)(1-2i)

$(1 + z i) (1 + 2 i) - (1 + z i) (1 - 2 i)$

Simplify each term.

Expand $(1 + z i) (1 + 2 i)$ using the FOIL Method.

Apply the distributive property.

$1 (1 + 2 i) + z i (1 + 2 i) - (1 + z i) (1 - 2 i)$

Apply the distributive property.

$1 \cdot 1 + 1 (2 i) + z i (1 + 2 i) - (1 + z i) (1 - 2 i)$

Apply the distributive property.

$1 \cdot 1 + 1 (2 i) + z i \cdot 1 + z i (2 i) - (1 + z i) (1 - 2 i)$

Simplify each term.

Multiply $1$ by $1$ .

$1 + 1 (2 i) + z i \cdot 1 + z i (2 i) - (1 + z i) (1 - 2 i)$

Multiply $2 i$ by $1$ .

$1 + 2 i + z i \cdot 1 + z i (2 i) - (1 + z i) (1 - 2 i)$

Multiply $z$ by $1$ .

$1 + 2 i + z i + z i (2 i) - (1 + z i) (1 - 2 i)$

Multiply $z i (2 i)$ .

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

$1 + 2 i + z i + z (2 (i^{1} i)) - (1 + z i) (1 - 2 i)$

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

$1 + 2 i + z i + z (2 (i^{1} i^{1})) - (1 + z i) (1 - 2 i)$

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

$1 + 2 i + z i + z (2 i^{1 + 1}) - (1 + z i) (1 - 2 i)$

Add $1$ and $1$ .

$1 + 2 i + z i + z (2 i^{2}) - (1 + z i) (1 - 2 i)$

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

$1 + 2 i + z i + z (2 \cdot - 1) - (1 + z i) (1 - 2 i)$

Multiply $2$ by $- 1$ .

$1 + 2 i + z i + z \cdot - 2 - (1 + z i) (1 - 2 i)$

Move $- 2$ to the left of $z$ .

$1 + 2 i + z i - 2 z - (1 + z i) (1 - 2 i)$

Apply the distributive property.

$1 + 2 i + z i - 2 z + (- 1 \cdot 1 - (z i)) (1 - 2 i)$

Multiply $- 1$ by $1$ .

$1 + 2 i + z i - 2 z + (- 1 - (z i)) (1 - 2 i)$

Expand $(- 1 - z i) (1 - 2 i)$ using the FOIL Method.

Apply the distributive property.

$1 + 2 i + z i - 2 z - 1 (1 - 2 i) - z i (1 - 2 i)$

Apply the distributive property.

$1 + 2 i + z i - 2 z - 1 \cdot 1 - 1 (- 2 i) - z i (1 - 2 i)$

Apply the distributive property.

$1 + 2 i + z i - 2 z - 1 \cdot 1 - 1 (- 2 i) - z i \cdot 1 - z i (- 2 i)$

Simplify each term.

Multiply $- 1$ by $1$ .

$1 + 2 i + z i - 2 z - 1 - 1 (- 2 i) - z i \cdot 1 - z i (- 2 i)$

Multiply $- 2$ by $- 1$ .

$1 + 2 i + z i - 2 z - 1 + 2 i - z i \cdot 1 - z i (- 2 i)$

Multiply $- 1$ by $1$ .

$1 + 2 i + z i - 2 z - 1 + 2 i - z i - z i (- 2 i)$

Multiply $- z i (- 2 i)$ .

Multiply $- 2$ by $- 1$ .

$1 + 2 i + z i - 2 z - 1 + 2 i - z i + 2 z i i$

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

$1 + 2 i + z i - 2 z - 1 + 2 i - z i + 2 z (i^{1} i)$

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

$1 + 2 i + z i - 2 z - 1 + 2 i - z i + 2 z (i^{1} i^{1})$

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

$1 + 2 i + z i - 2 z - 1 + 2 i - z i + 2 z i^{1 + 1}$

Add $1$ and $1$ .

$1 + 2 i + z i - 2 z - 1 + 2 i - z i + 2 z i^{2}$

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

$1 + 2 i + z i - 2 z - 1 + 2 i - z i + 2 z \cdot - 1$

Multiply $- 1$ by $2$ .

$1 + 2 i + z i - 2 z - 1 + 2 i - z i - 2 z$

Simplify by adding terms.

Combine the opposite terms in $1 + 2 i + z i - 2 z - 1 + 2 i - z i - 2 z$ .

Subtract $1$ from $1$ .

$2 i + z i - 2 z + 0 + 2 i - z i - 2 z$

Add $2 i + z i - 2 z$ and $0$ .

$2 i + z i - 2 z + 2 i - z i - 2 z$

Subtract $z i$ from $z i$ .

$2 i + 0 - 2 z + 2 i - 2 z$

Add $2 i$ and $0$ .

$2 i - 2 z + 2 i - 2 z$

Add $2 i$ and $2 i$ .

$- 2 z + 4 i - 2 z$

Subtract $2 z$ from $- 2 z$ .

$- 4 z + 4 i$

Simplify (1+zi)(1+2i)-(1+zi)(1-2i)

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