Simplify (-2i)^10

${(- 2 i)}^{10}$

Step 1

Apply the product rule to $- 2 i$ .

${(- 2)}^{10} i^{10}$

Step 2

Raise $- 2$ to the power of $10$ .

$1024 i^{10}$

Step 3

Rewrite $i^{10}$ as ${(i^{4})}^{2} i^{2}$ .

Factor out $i^{8}$ .

$1024 (i^{8} i^{2})$

Rewrite $i^{8}$ as ${(i^{4})}^{2}$ .

$1024 ({(i^{4})}^{2} i^{2})$

$1024 ({(i^{4})}^{2} i^{2})$

Step 4

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

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

$1024 ({({(i^{2})}^{2})}^{2} i^{2})$

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

$1024 ({({(- 1)}^{2})}^{2} i^{2})$

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

$1024 (1^{2} i^{2})$

$1024 (1^{2} i^{2})$

Step 5

One to any power is one.

$1024 (1 i^{2})$

Step 6

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

$1024 i^{2}$

Step 7

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

$1024 \cdot - 1$

Step 8

Multiply $1024$ by $- 1$ .

$- 1024$

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