Simplify (6+5yi)(1+4i)+(1-5yi)(1+4i)

$(6 + 5 y i) (1 + 4 i) + (1 - 5 y i) (1 + 4 i)$

Simplify each term.

Expand $(6 + 5 y i) (1 + 4 i)$ using the FOIL Method.

Apply the distributive property.

$6 (1 + 4 i) + 5 y i (1 + 4 i) + (1 - 5 y i) (1 + 4 i)$

Apply the distributive property.

$6 \cdot 1 + 6 (4 i) + 5 y i (1 + 4 i) + (1 - 5 y i) (1 + 4 i)$

Apply the distributive property.

$6 \cdot 1 + 6 (4 i) + 5 y i \cdot 1 + 5 y i (4 i) + (1 - 5 y i) (1 + 4 i)$

Simplify each term.

Multiply $6$ by $1$ .

$6 + 6 (4 i) + 5 y i \cdot 1 + 5 y i (4 i) + (1 - 5 y i) (1 + 4 i)$

Multiply $4$ by $6$ .

$6 + 24 i + 5 y i \cdot 1 + 5 y i (4 i) + (1 - 5 y i) (1 + 4 i)$

Multiply $5$ by $1$ .

$6 + 24 i + 5 y i + 5 y i (4 i) + (1 - 5 y i) (1 + 4 i)$

Multiply $5 y i (4 i)$ .

Multiply $4$ by $5$ .

$6 + 24 i + 5 y i + 20 y i i + (1 - 5 y i) (1 + 4 i)$

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

$6 + 24 i + 5 y i + 20 y (i^{1} i) + (1 - 5 y i) (1 + 4 i)$

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

$6 + 24 i + 5 y i + 20 y (i^{1} i^{1}) + (1 - 5 y i) (1 + 4 i)$

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

$6 + 24 i + 5 y i + 20 y i^{1 + 1} + (1 - 5 y i) (1 + 4 i)$

Add $1$ and $1$ .

$6 + 24 i + 5 y i + 20 y i^{2} + (1 - 5 y i) (1 + 4 i)$

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

$6 + 24 i + 5 y i + 20 y \cdot - 1 + (1 - 5 y i) (1 + 4 i)$

Multiply $- 1$ by $20$ .

$6 + 24 i + 5 y i - 20 y + (1 - 5 y i) (1 + 4 i)$

Expand $(1 - 5 y i) (1 + 4 i)$ using the FOIL Method.

Apply the distributive property.

$6 + 24 i + 5 y i - 20 y + 1 (1 + 4 i) - 5 y i (1 + 4 i)$

Apply the distributive property.

$6 + 24 i + 5 y i - 20 y + 1 \cdot 1 + 1 (4 i) - 5 y i (1 + 4 i)$

Apply the distributive property.

$6 + 24 i + 5 y i - 20 y + 1 \cdot 1 + 1 (4 i) - 5 y i \cdot 1 - 5 y i (4 i)$

Simplify each term.

Multiply $1$ by $1$ .

$6 + 24 i + 5 y i - 20 y + 1 + 1 (4 i) - 5 y i \cdot 1 - 5 y i (4 i)$

Multiply $4 i$ by $1$ .

$6 + 24 i + 5 y i - 20 y + 1 + 4 i - 5 y i \cdot 1 - 5 y i (4 i)$

Multiply $- 5$ by $1$ .

$6 + 24 i + 5 y i - 20 y + 1 + 4 i - 5 y i - 5 y i (4 i)$

Multiply $- 5 y i (4 i)$ .

Multiply $4$ by $- 5$ .

$6 + 24 i + 5 y i - 20 y + 1 + 4 i - 5 y i - 20 y i i$

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

$6 + 24 i + 5 y i - 20 y + 1 + 4 i - 5 y i - 20 y (i^{1} i)$

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

$6 + 24 i + 5 y i - 20 y + 1 + 4 i - 5 y i - 20 y (i^{1} i^{1})$

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

$6 + 24 i + 5 y i - 20 y + 1 + 4 i - 5 y i - 20 y i^{1 + 1}$

Add $1$ and $1$ .

$6 + 24 i + 5 y i - 20 y + 1 + 4 i - 5 y i - 20 y i^{2}$

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

$6 + 24 i + 5 y i - 20 y + 1 + 4 i - 5 y i - 20 y \cdot - 1$

Multiply $- 1$ by $- 20$ .

$6 + 24 i + 5 y i - 20 y + 1 + 4 i - 5 y i + 20 y$

Simplify by adding terms.

Combine the opposite terms in $6 + 24 i + 5 y i - 20 y + 1 + 4 i - 5 y i + 20 y$ .

Subtract $5 y i$ from $5 y i$ .

$6 + 24 i + 0 - 20 y + 1 + 4 i + 20 y$

Add $6 + 24 i$ and $0$ .

$6 + 24 i - 20 y + 1 + 4 i + 20 y$

Add $- 20 y$ and $20 y$ .

$6 + 24 i + 0 + 1 + 4 i$

Add $6 + 24 i$ and $0$ .

$6 + 24 i + 1 + 4 i$

Add $6$ and $1$ .

$7 + 24 i + 4 i$

Add $24 i$ and $4 i$ .

$7 + 28 i$

Simplify (6+5yi)(1+4i)+(1-5yi)(1+4i)

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