Simplify (5+2yi)(4-3i)-(5-2yi)(4-3i)

$(5 + 2 y i) (4 - 3 i) - (5 - 2 y i) (4 - 3 i)$

Simplify each term.

Expand $(5 + 2 y i) (4 - 3 i)$ using the FOIL Method.

Apply the distributive property.

$5 (4 - 3 i) + 2 y i (4 - 3 i) - (5 - 2 y i) (4 - 3 i)$

Apply the distributive property.

$5 \cdot 4 + 5 (- 3 i) + 2 y i (4 - 3 i) - (5 - 2 y i) (4 - 3 i)$

Apply the distributive property.

$5 \cdot 4 + 5 (- 3 i) + 2 y i \cdot 4 + 2 y i (- 3 i) - (5 - 2 y i) (4 - 3 i)$

Simplify each term.

Multiply $5$ by $4$ .

$20 + 5 (- 3 i) + 2 y i \cdot 4 + 2 y i (- 3 i) - (5 - 2 y i) (4 - 3 i)$

Multiply $- 3$ by $5$ .

$20 - 15 i + 2 y i \cdot 4 + 2 y i (- 3 i) - (5 - 2 y i) (4 - 3 i)$

Multiply $4$ by $2$ .

$20 - 15 i + 8 y i + 2 y i (- 3 i) - (5 - 2 y i) (4 - 3 i)$

Multiply $2 y i (- 3 i)$ .

Multiply $- 3$ by $2$ .

$20 - 15 i + 8 y i - 6 y i i - (5 - 2 y i) (4 - 3 i)$

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

$20 - 15 i + 8 y i - 6 y (i^{1} i) - (5 - 2 y i) (4 - 3 i)$

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

$20 - 15 i + 8 y i - 6 y (i^{1} i^{1}) - (5 - 2 y i) (4 - 3 i)$

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

$20 - 15 i + 8 y i - 6 y i^{1 + 1} - (5 - 2 y i) (4 - 3 i)$

Add $1$ and $1$ .

$20 - 15 i + 8 y i - 6 y i^{2} - (5 - 2 y i) (4 - 3 i)$

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

$20 - 15 i + 8 y i - 6 y \cdot - 1 - (5 - 2 y i) (4 - 3 i)$

Multiply $- 1$ by $- 6$ .

$20 - 15 i + 8 y i + 6 y - (5 - 2 y i) (4 - 3 i)$

Apply the distributive property.

$20 - 15 i + 8 y i + 6 y + (- 1 \cdot 5 - (- 2 y i)) (4 - 3 i)$

Multiply $- 1$ by $5$ .

$20 - 15 i + 8 y i + 6 y + (- 5 - (- 2 y i)) (4 - 3 i)$

Multiply $- 2$ by $- 1$ .

$20 - 15 i + 8 y i + 6 y + (- 5 + 2 (y i)) (4 - 3 i)$

Expand $(- 5 + 2 y i) (4 - 3 i)$ using the FOIL Method.

Apply the distributive property.

$20 - 15 i + 8 y i + 6 y - 5 (4 - 3 i) + 2 y i (4 - 3 i)$

Apply the distributive property.

$20 - 15 i + 8 y i + 6 y - 5 \cdot 4 - 5 (- 3 i) + 2 y i (4 - 3 i)$

Apply the distributive property.

$20 - 15 i + 8 y i + 6 y - 5 \cdot 4 - 5 (- 3 i) + 2 y i \cdot 4 + 2 y i (- 3 i)$

Simplify each term.

Multiply $- 5$ by $4$ .

$20 - 15 i + 8 y i + 6 y - 20 - 5 (- 3 i) + 2 y i \cdot 4 + 2 y i (- 3 i)$

Multiply $- 3$ by $- 5$ .

$20 - 15 i + 8 y i + 6 y - 20 + 15 i + 2 y i \cdot 4 + 2 y i (- 3 i)$

Multiply $4$ by $2$ .

$20 - 15 i + 8 y i + 6 y - 20 + 15 i + 8 y i + 2 y i (- 3 i)$

Multiply $2 y i (- 3 i)$ .

Multiply $- 3$ by $2$ .

$20 - 15 i + 8 y i + 6 y - 20 + 15 i + 8 y i - 6 y i i$

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

$20 - 15 i + 8 y i + 6 y - 20 + 15 i + 8 y i - 6 y (i^{1} i)$

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

$20 - 15 i + 8 y i + 6 y - 20 + 15 i + 8 y i - 6 y (i^{1} i^{1})$

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

$20 - 15 i + 8 y i + 6 y - 20 + 15 i + 8 y i - 6 y i^{1 + 1}$

Add $1$ and $1$ .

$20 - 15 i + 8 y i + 6 y - 20 + 15 i + 8 y i - 6 y i^{2}$

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

$20 - 15 i + 8 y i + 6 y - 20 + 15 i + 8 y i - 6 y \cdot - 1$

Multiply $- 1$ by $- 6$ .

$20 - 15 i + 8 y i + 6 y - 20 + 15 i + 8 y i + 6 y$

Simplify by adding terms.

Combine the opposite terms in $20 - 15 i + 8 y i + 6 y - 20 + 15 i + 8 y i + 6 y$ .

Subtract $20$ from $20$ .

$- 15 i + 8 y i + 6 y + 0 + 15 i + 8 y i + 6 y$

Add $- 15 i + 8 y i + 6 y$ and $0$ .

$- 15 i + 8 y i + 6 y + 15 i + 8 y i + 6 y$

Add $- 15 i$ and $15 i$ .

$8 y i + 6 y + 0 + 8 y i + 6 y$

Add $8 y i + 6 y$ and $0$ .

$8 y i + 6 y + 8 y i + 6 y$

Add $8 y i$ and $8 y i$ .

$16 y i + 6 y + 6 y$

Add $6 y$ and $6 y$ .

$16 y i + 12 y$

Simplify (5+2yi)(4-3i)-(5-2yi)(4-3i)

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