Simplify (4+2i)^2

${(4 + 2 i)}^{2}$

Step 1

Rewrite ${(4 + 2 i)}^{2}$ as $(4 + 2 i) (4 + 2 i)$ .

$(4 + 2 i) (4 + 2 i)$

Step 2

Expand $(4 + 2 i) (4 + 2 i)$ using the FOIL Method.

Apply the distributive property.

$4 (4 + 2 i) + 2 i (4 + 2 i)$

Apply the distributive property.

$4 \cdot 4 + 4 (2 i) + 2 i (4 + 2 i)$

Apply the distributive property.

$4 \cdot 4 + 4 (2 i) + 2 i \cdot 4 + 2 i (2 i)$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$16 + 4 (2 i) + 2 i \cdot 4 + 2 i (2 i)$

Multiply $2$ by $4$ .

$16 + 8 i + 2 i \cdot 4 + 2 i (2 i)$

Multiply $4$ by $2$ .

$16 + 8 i + 8 i + 2 i (2 i)$

Multiply $2 i (2 i)$ .

Multiply $2$ by $2$ .

$16 + 8 i + 8 i + 4 i i$

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

$16 + 8 i + 8 i + 4 (i^{1} i)$

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

$16 + 8 i + 8 i + 4 (i^{1} i^{1})$

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

$16 + 8 i + 8 i + 4 i^{1 + 1}$

Add $1$ and $1$ .

$16 + 8 i + 8 i + 4 i^{2}$

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

$16 + 8 i + 8 i + 4 \cdot - 1$

Multiply $4$ by $- 1$ .

$16 + 8 i + 8 i - 4$

Subtract $4$ from $16$ .

$12 + 8 i + 8 i$

Add $8 i$ and $8 i$ .

$12 + 16 i$