Simplify (2-5i)(1-4i)

$(2 - 5 i) (1 - 4 i)$

Step 1

Expand $(2 - 5 i) (1 - 4 i)$ using the FOIL Method.

Apply the distributive property.

$2 (1 - 4 i) - 5 i (1 - 4 i)$

Apply the distributive property.

$2 \cdot 1 + 2 (- 4 i) - 5 i (1 - 4 i)$

Apply the distributive property.

$2 \cdot 1 + 2 (- 4 i) - 5 i \cdot 1 - 5 i (- 4 i)$

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $2$ by $1$ .

$2 + 2 (- 4 i) - 5 i \cdot 1 - 5 i (- 4 i)$

Multiply $- 4$ by $2$ .

$2 - 8 i - 5 i \cdot 1 - 5 i (- 4 i)$

Multiply $- 5$ by $1$ .

$2 - 8 i - 5 i - 5 i (- 4 i)$

Multiply $- 5 i (- 4 i)$ .

Multiply $- 4$ by $- 5$ .

$2 - 8 i - 5 i + 20 i i$

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

$2 - 8 i - 5 i + 20 (i^{1} i)$

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

$2 - 8 i - 5 i + 20 (i^{1} i^{1})$

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

$2 - 8 i - 5 i + 20 i^{1 + 1}$

Add $1$ and $1$ .

$2 - 8 i - 5 i + 20 i^{2}$

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

$2 - 8 i - 5 i + 20 \cdot - 1$

Multiply $20$ by $- 1$ .

$2 - 8 i - 5 i - 20$

Subtract $20$ from $2$ .

$- 18 - 8 i - 5 i$

Subtract $5 i$ from $- 8 i$ .

$- 18 - 13 i$