Simplify (-2+2i)(-5i)^3

$(- 2 + 2 i) {(- 5 i)}^{3}$

Step 1

Apply the product rule to $- 5 i$ .

$(- 2 + 2 i) ({(- 5)}^{3} i^{3})$

Step 2

Raise $- 5$ to the power of $3$ .

$(- 2 + 2 i) (- 125 i^{3})$

Step 3

Factor out $i^{2}$ .

$(- 2 + 2 i) (- 125 (i^{2} \cdot i))$

Step 4

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

$(- 2 + 2 i) (- 125 (- 1 \cdot i))$

Step 5

Simplify the expression.

Rewrite $- 1 i$ as $- i$ .

$(- 2 + 2 i) (- 125 (- i))$

Multiply $- 1$ by $- 125$ .

$(- 2 + 2 i) (125 i)$

$(- 2 + 2 i) (125 i)$

Step 6

Apply the distributive property.

$- 2 (125 i) + 2 i (125 i)$

Step 7

Multiply $125$ by $- 2$ .

$- 250 i + 2 i (125 i)$

Step 8

Multiply $2 i (125 i)$ .

Multiply $125$ by $2$ .

$- 250 i + 250 i i$

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

$- 250 i + 250 (i^{1} i)$

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

$- 250 i + 250 (i^{1} i^{1})$

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

$- 250 i + 250 i^{1 + 1}$

Add $1$ and $1$ .

$- 250 i + 250 i^{2}$

$- 250 i + 250 i^{2}$

Step 9

Simplify each term.

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

$- 250 i + 250 \cdot - 1$

Multiply $250$ by $- 1$ .

$- 250 i - 250$

$- 250 i - 250$

Step 10

Reorder $- 250 i$ and $- 250$ .

$- 250 - 250 i$

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