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

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

Step 1

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Simplify each term.

Multiply $5$ by $1$ .

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

Multiply $- 2 i$ by $1$ .

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

Multiply $5$ by $- 4$ .

$5 - 2 i - 20 i - 4 i (- 2 i)$

Multiply $- 4 i (- 2 i)$ .

Multiply $- 2$ by $- 4$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

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

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

Multiply $8$ by $- 1$ .

$5 - 2 i - 20 i - 8$

Subtract $8$ from $5$ .

$- 3 - 2 i - 20 i$

Subtract $20 i$ from $- 2 i$ .

$- 3 - 22 i$