Evaluate i^7*i^17

$i^{7} \cdot i^{17}$

Step 1

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

Factor out $i^{4}$ .

$i^{4} i^{3} \cdot i^{17}$

Factor out $i^{2}$ .

$i^{4} (i^{2} \cdot i) \cdot i^{17}$

$i^{4} (i^{2} \cdot i) \cdot i^{17}$

Step 2

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

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

${(i^{2})}^{2} (i^{2} \cdot i) \cdot i^{17}$

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

${(- 1)}^{2} (i^{2} \cdot i) \cdot i^{17}$

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

$1 (i^{2} \cdot i) \cdot i^{17}$

$1 (i^{2} \cdot i) \cdot i^{17}$

Step 3

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

$i^{2} \cdot i \cdot i^{17}$

Step 4

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

$- 1 \cdot i \cdot i^{17}$

Step 5

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

$- i \cdot i^{17}$

Step 6

Rewrite $i^{17}$ as ${(i^{4})}^{4} i$ .

Factor out $i^{16}$ .

$- i \cdot (i^{16} i)$

Rewrite $i^{16}$ as ${(i^{4})}^{4}$ .

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

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

Step 7

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

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

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

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

$- i \cdot ({({(- 1)}^{2})}^{4} i)$

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

$- i \cdot (1^{4} i)$

$- i \cdot (1^{4} i)$

Step 8

One to any power is one.

$- i \cdot (1 i)$

Step 9

Multiply $i$ by $1$ .

$- i \cdot i$

Step 10

Multiply $- i i$ .

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

$- (i^{1} i)$

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

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

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

$- i^{1 + 1}$

Add $1$ and $1$ .

$- i^{2}$

$- i^{2}$

Step 11

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

$- - 1$

Step 12

Multiply $- 1$ by $- 1$ .

$1$

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