Evaluate (6+7i)^2

${(6 + 7 i)}^{2}$

Step 1

Rewrite ${(6 + 7 i)}^{2}$ as $(6 + 7 i) (6 + 7 i)$ .

$(6 + 7 i) (6 + 7 i)$

Step 2

Expand $(6 + 7 i) (6 + 7 i)$ using the FOIL Method.

Apply the distributive property.

$6 (6 + 7 i) + 7 i (6 + 7 i)$

Apply the distributive property.

$6 \cdot 6 + 6 (7 i) + 7 i (6 + 7 i)$

Apply the distributive property.

$6 \cdot 6 + 6 (7 i) + 7 i \cdot 6 + 7 i (7 i)$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

$36 + 6 (7 i) + 7 i \cdot 6 + 7 i (7 i)$

Multiply $7$ by $6$ .

$36 + 42 i + 7 i \cdot 6 + 7 i (7 i)$

Multiply $6$ by $7$ .

$36 + 42 i + 42 i + 7 i (7 i)$

Multiply $7 i (7 i)$ .

Multiply $7$ by $7$ .

$36 + 42 i + 42 i + 49 i i$

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

$36 + 42 i + 42 i + 49 (i^{1} i)$

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

$36 + 42 i + 42 i + 49 (i^{1} i^{1})$

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

$36 + 42 i + 42 i + 49 i^{1 + 1}$

Add $1$ and $1$ .

$36 + 42 i + 42 i + 49 i^{2}$

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

$36 + 42 i + 42 i + 49 \cdot - 1$

Multiply $49$ by $- 1$ .

$36 + 42 i + 42 i - 49$

Subtract $49$ from $36$ .

$- 13 + 42 i + 42 i$

Add $42 i$ and $42 i$ .

$- 13 + 84 i$