Simplify (4+7i)(3+4i)

$(4 + 7 i) (3 + 4 i)$

Step 1

Expand $(4 + 7 i) (3 + 4 i)$ using the FOIL Method.

Apply the distributive property.

$4 (3 + 4 i) + 7 i (3 + 4 i)$

Apply the distributive property.

$4 \cdot 3 + 4 (4 i) + 7 i (3 + 4 i)$

Apply the distributive property.

$4 \cdot 3 + 4 (4 i) + 7 i \cdot 3 + 7 i (4 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $3$ .

$12 + 4 (4 i) + 7 i \cdot 3 + 7 i (4 i)$

Multiply $4$ by $4$ .

$12 + 16 i + 7 i \cdot 3 + 7 i (4 i)$

Multiply $3$ by $7$ .

$12 + 16 i + 21 i + 7 i (4 i)$

Multiply $7 i (4 i)$ .

Multiply $4$ by $7$ .

$12 + 16 i + 21 i + 28 i i$

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

$12 + 16 i + 21 i + 28 (i^{1} i)$

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

$12 + 16 i + 21 i + 28 (i^{1} i^{1})$

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

$12 + 16 i + 21 i + 28 i^{1 + 1}$

Add $1$ and $1$ .

$12 + 16 i + 21 i + 28 i^{2}$

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

$12 + 16 i + 21 i + 28 \cdot - 1$

Multiply $28$ by $- 1$ .

$12 + 16 i + 21 i - 28$

Subtract $28$ from $12$ .

$- 16 + 16 i + 21 i$

Add $16 i$ and $21 i$ .

$- 16 + 37 i$