Simplify (4-5i)^2

${(4 - 5 i)}^{2}$

Step 1

Rewrite ${(4 - 5 i)}^{2}$ as $(4 - 5 i) (4 - 5 i)$ .

$(4 - 5 i) (4 - 5 i)$

Step 2

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

Apply the distributive property.

$4 (4 - 5 i) - 5 i (4 - 5 i)$

Apply the distributive property.

$4 \cdot 4 + 4 (- 5 i) - 5 i (4 - 5 i)$

Apply the distributive property.

$4 \cdot 4 + 4 (- 5 i) - 5 i \cdot 4 - 5 i (- 5 i)$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $4$ by $4$ .

$16 + 4 (- 5 i) - 5 i \cdot 4 - 5 i (- 5 i)$

Multiply $- 5$ by $4$ .

$16 - 20 i - 5 i \cdot 4 - 5 i (- 5 i)$

Multiply $4$ by $- 5$ .

$16 - 20 i - 20 i - 5 i (- 5 i)$

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

Multiply $- 5$ by $- 5$ .

$16 - 20 i - 20 i + 25 i i$

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

$16 - 20 i - 20 i + 25 (i^{1} i)$

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

$16 - 20 i - 20 i + 25 (i^{1} i^{1})$

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

$16 - 20 i - 20 i + 25 i^{1 + 1}$

Add $1$ and $1$ .

$16 - 20 i - 20 i + 25 i^{2}$

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

$16 - 20 i - 20 i + 25 \cdot - 1$

Multiply $25$ by $- 1$ .

$16 - 20 i - 20 i - 25$

Subtract $25$ from $16$ .

$- 9 - 20 i - 20 i$

Subtract $20 i$ from $- 20 i$ .

$- 9 - 40 i$