Simplify (5-2i)^2

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

Step 1

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

$(5 - 2 i) (5 - 2 i)$

Step 2

Expand $(5 - 2 i) (5 - 2 i)$ using the FOIL Method.

Apply the distributive property.

$5 (5 - 2 i) - 2 i (5 - 2 i)$

Apply the distributive property.

$5 \cdot 5 + 5 (- 2 i) - 2 i (5 - 2 i)$

Apply the distributive property.

$5 \cdot 5 + 5 (- 2 i) - 2 i \cdot 5 - 2 i (- 2 i)$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $5$ .

$25 + 5 (- 2 i) - 2 i \cdot 5 - 2 i (- 2 i)$

Multiply $- 2$ by $5$ .

$25 - 10 i - 2 i \cdot 5 - 2 i (- 2 i)$

Multiply $5$ by $- 2$ .

$25 - 10 i - 10 i - 2 i (- 2 i)$

Multiply $- 2 i (- 2 i)$ .

Multiply $- 2$ by $- 2$ .

$25 - 10 i - 10 i + 4 i i$

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

$25 - 10 i - 10 i + 4 (i^{1} i)$

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

$25 - 10 i - 10 i + 4 (i^{1} i^{1})$

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

$25 - 10 i - 10 i + 4 i^{1 + 1}$

Add $1$ and $1$ .

$25 - 10 i - 10 i + 4 i^{2}$

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

$25 - 10 i - 10 i + 4 \cdot - 1$

Multiply $4$ by $- 1$ .

$25 - 10 i - 10 i - 4$

Subtract $4$ from $25$ .

$21 - 10 i - 10 i$

Subtract $10 i$ from $- 10 i$ .

$21 - 20 i$