Simplify (-i)^7

${(- i)}^{7}$

Step 1

Apply the product rule to $- i$ .

${(- 1)}^{7} i^{7}$

Step 2

Raise $- 1$ to the power of $7$ .

$- i^{7}$

Step 3

Rewrite $i^{7}$ as $i^{4} (i^{2} \cdot i)$ .

Factor out $i^{4}$ .

$- (i^{4} i^{3})$

Factor out $i^{2}$ .

$- (i^{4} (i^{2} \cdot i))$

$- (i^{4} (i^{2} \cdot i))$

Step 4

Rewrite $i^{4}$ as $1$ .

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

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

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

$- ({(- 1)}^{2} (i^{2} \cdot i))$

Raise $- 1$ to the power of $2$ .

$- (1 (i^{2} \cdot i))$

$- (1 (i^{2} \cdot i))$

Step 5

Multiply $i^{2} \cdot i$ by $1$ .

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

Step 6

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

$- (- 1 \cdot i)$

Step 7

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

$- - i$

Step 8

Multiply $- 1$ by $- 1$ .

$1 i$

Step 9

Multiply $i$ by $1$ .

$i$

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