Simplify (3+5zi)(1+2i)-(2+5zi)(1+2i)

$(3 + 5 z i) (1 + 2 i) - (2 + 5 z i) (1 + 2 i)$

Simplify each term.

Expand $(3 + 5 z i) (1 + 2 i)$ using the FOIL Method.

Apply the distributive property.

$3 (1 + 2 i) + 5 z i (1 + 2 i) - (2 + 5 z i) (1 + 2 i)$

Apply the distributive property.

$3 \cdot 1 + 3 (2 i) + 5 z i (1 + 2 i) - (2 + 5 z i) (1 + 2 i)$

Apply the distributive property.

$3 \cdot 1 + 3 (2 i) + 5 z i \cdot 1 + 5 z i (2 i) - (2 + 5 z i) (1 + 2 i)$

Simplify each term.

Multiply $3$ by $1$ .

$3 + 3 (2 i) + 5 z i \cdot 1 + 5 z i (2 i) - (2 + 5 z i) (1 + 2 i)$

Multiply $2$ by $3$ .

$3 + 6 i + 5 z i \cdot 1 + 5 z i (2 i) - (2 + 5 z i) (1 + 2 i)$

Multiply $5$ by $1$ .

$3 + 6 i + 5 z i + 5 z i (2 i) - (2 + 5 z i) (1 + 2 i)$

Multiply $5 z i (2 i)$ .

Multiply $2$ by $5$ .

$3 + 6 i + 5 z i + 10 z i i - (2 + 5 z i) (1 + 2 i)$

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

$3 + 6 i + 5 z i + 10 z (i^{1} i) - (2 + 5 z i) (1 + 2 i)$

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

$3 + 6 i + 5 z i + 10 z (i^{1} i^{1}) - (2 + 5 z i) (1 + 2 i)$

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

$3 + 6 i + 5 z i + 10 z i^{1 + 1} - (2 + 5 z i) (1 + 2 i)$

Add $1$ and $1$ .

$3 + 6 i + 5 z i + 10 z i^{2} - (2 + 5 z i) (1 + 2 i)$

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

$3 + 6 i + 5 z i + 10 z \cdot - 1 - (2 + 5 z i) (1 + 2 i)$

Multiply $- 1$ by $10$ .

$3 + 6 i + 5 z i - 10 z - (2 + 5 z i) (1 + 2 i)$

Apply the distributive property.

$3 + 6 i + 5 z i - 10 z + (- 1 \cdot 2 - (5 z i)) (1 + 2 i)$

Multiply $- 1$ by $2$ .

$3 + 6 i + 5 z i - 10 z + (- 2 - (5 z i)) (1 + 2 i)$

Multiply $5$ by $- 1$ .

$3 + 6 i + 5 z i - 10 z + (- 2 - 5 (z i)) (1 + 2 i)$

Expand $(- 2 - 5 z i) (1 + 2 i)$ using the FOIL Method.

Apply the distributive property.

$3 + 6 i + 5 z i - 10 z - 2 (1 + 2 i) - 5 z i (1 + 2 i)$

Apply the distributive property.

$3 + 6 i + 5 z i - 10 z - 2 \cdot 1 - 2 (2 i) - 5 z i (1 + 2 i)$

Apply the distributive property.

$3 + 6 i + 5 z i - 10 z - 2 \cdot 1 - 2 (2 i) - 5 z i \cdot 1 - 5 z i (2 i)$

Simplify each term.

Multiply $- 2$ by $1$ .

$3 + 6 i + 5 z i - 10 z - 2 - 2 (2 i) - 5 z i \cdot 1 - 5 z i (2 i)$

Multiply $2$ by $- 2$ .

$3 + 6 i + 5 z i - 10 z - 2 - 4 i - 5 z i \cdot 1 - 5 z i (2 i)$

Multiply $- 5$ by $1$ .

$3 + 6 i + 5 z i - 10 z - 2 - 4 i - 5 z i - 5 z i (2 i)$

Multiply $- 5 z i (2 i)$ .

Multiply $2$ by $- 5$ .

$3 + 6 i + 5 z i - 10 z - 2 - 4 i - 5 z i - 10 z i i$

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

$3 + 6 i + 5 z i - 10 z - 2 - 4 i - 5 z i - 10 z (i^{1} i)$

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

$3 + 6 i + 5 z i - 10 z - 2 - 4 i - 5 z i - 10 z (i^{1} i^{1})$

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

$3 + 6 i + 5 z i - 10 z - 2 - 4 i - 5 z i - 10 z i^{1 + 1}$

Add $1$ and $1$ .

$3 + 6 i + 5 z i - 10 z - 2 - 4 i - 5 z i - 10 z i^{2}$

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

$3 + 6 i + 5 z i - 10 z - 2 - 4 i - 5 z i - 10 z \cdot - 1$

Multiply $- 1$ by $- 10$ .

$3 + 6 i + 5 z i - 10 z - 2 - 4 i - 5 z i + 10 z$

Simplify by adding terms.

Combine the opposite terms in $3 + 6 i + 5 z i - 10 z - 2 - 4 i - 5 z i + 10 z$ .

Subtract $5 z i$ from $5 z i$ .

$3 + 6 i + 0 - 10 z - 2 - 4 i + 10 z$

Add $3 + 6 i$ and $0$ .

$3 + 6 i - 10 z - 2 - 4 i + 10 z$

Add $- 10 z$ and $10 z$ .

$3 + 6 i + 0 - 2 - 4 i$

Add $3 + 6 i$ and $0$ .

$3 + 6 i - 2 - 4 i$

Subtract $2$ from $3$ .

$1 + 6 i - 4 i$

Subtract $4 i$ from $6 i$ .

$1 + 2 i$

Simplify (3+5zi)(1+2i)-(2+5zi)(1+2i)

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