Simplify (7-4i)^2

${(7 - 4 i)}^{2}$

Step 1

Rewrite ${(7 - 4 i)}^{2}$ as $(7 - 4 i) (7 - 4 i)$ .

$(7 - 4 i) (7 - 4 i)$

Step 2

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

Apply the distributive property.

$7 (7 - 4 i) - 4 i (7 - 4 i)$

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $7$ by $7$ .

$49 + 7 (- 4 i) - 4 i \cdot 7 - 4 i (- 4 i)$

Multiply $- 4$ by $7$ .

$49 - 28 i - 4 i \cdot 7 - 4 i (- 4 i)$

Multiply $7$ by $- 4$ .

$49 - 28 i - 28 i - 4 i (- 4 i)$

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

Multiply $- 4$ by $- 4$ .

$49 - 28 i - 28 i + 16 i i$

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

$49 - 28 i - 28 i + 16 (i^{1} i)$

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

$49 - 28 i - 28 i + 16 (i^{1} i^{1})$

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

$49 - 28 i - 28 i + 16 i^{1 + 1}$

Add $1$ and $1$ .

$49 - 28 i - 28 i + 16 i^{2}$

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

$49 - 28 i - 28 i + 16 \cdot - 1$

Multiply $16$ by $- 1$ .

$49 - 28 i - 28 i - 16$

Subtract $16$ from $49$ .

$33 - 28 i - 28 i$

Subtract $28 i$ from $- 28 i$ .

$33 - 56 i$