Algebra Examples

$i^{19} \cdot i^{9} \cdot i^{6} \cdot i^{10}$

Step 1

Rewrite $i^{19}$ as ${(i^{4})}^{4} (i^{2} \cdot i)$ .

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Step 1.1

Factor out $i^{16}$ .

$i^{16} i^{3} \cdot i^{9} \cdot i^{6} \cdot i^{10}$

Step 1.2

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

${(i^{4})}^{4} i^{3} \cdot i^{9} \cdot i^{6} \cdot i^{10}$

Step 1.3

Factor out $i^{2}$ .

${(i^{4})}^{4} (i^{2} \cdot i) \cdot i^{9} \cdot i^{6} \cdot i^{10}$

Step 2

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

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Step 2.1

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

${({(i^{2})}^{2})}^{4} (i^{2} \cdot i) \cdot i^{9} \cdot i^{6} \cdot i^{10}$

Step 2.2

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

${({(- 1)}^{2})}^{4} (i^{2} \cdot i) \cdot i^{9} \cdot i^{6} \cdot i^{10}$

Step 2.3

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

$1^{4} (i^{2} \cdot i) \cdot i^{9} \cdot i^{6} \cdot i^{10}$

Step 3

One to any power is one.

$1 (i^{2} \cdot i) \cdot i^{9} \cdot i^{6} \cdot i^{10}$

Step 4

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

$i^{2} \cdot i \cdot i^{9} \cdot i^{6} \cdot i^{10}$

Step 5

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

$- 1 \cdot i \cdot i^{9} \cdot i^{6} \cdot i^{10}$

Step 6

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

$- i \cdot i^{9} \cdot i^{6} \cdot i^{10}$

Step 7

Rewrite $i^{9}$ as ${(i^{4})}^{2} i$ .

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Step 7.1

Factor out $i^{8}$ .

$- i \cdot (i^{8} i) \cdot i^{6} \cdot i^{10}$

Step 7.2

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

$- i \cdot ({(i^{4})}^{2} i) \cdot i^{6} \cdot i^{10}$

Step 8

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

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Step 8.1

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

$- i \cdot ({({(i^{2})}^{2})}^{2} i) \cdot i^{6} \cdot i^{10}$

Step 8.2

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

$- i \cdot ({({(- 1)}^{2})}^{2} i) \cdot i^{6} \cdot i^{10}$

Step 8.3

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

$- i \cdot (1^{2} i) \cdot i^{6} \cdot i^{10}$

Step 9

One to any power is one.

$- i \cdot (1 i) \cdot i^{6} \cdot i^{10}$

Step 10

Multiply $i$ by $1$ .

$- i \cdot i \cdot i^{6} \cdot i^{10}$

Step 11

Multiply $- i i$ .

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Step 11.1

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

$- (i^{1} i) \cdot i^{6} \cdot i^{10}$

Step 11.2

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

$- (i^{1} i^{1}) \cdot i^{6} \cdot i^{10}$

Step 11.3

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

$- i^{1 + 1} \cdot i^{6} \cdot i^{10}$

Step 11.4

Add $1$ and $1$ .

$- i^{2} \cdot i^{6} \cdot i^{10}$

Step 12

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

$- - 1 \cdot i^{6} \cdot i^{10}$

Step 13

Multiply $- 1$ by $- 1$ .

$1 \cdot i^{6} \cdot i^{10}$

Step 14

Multiply $i^{6}$ by $1$ .

$i^{6} \cdot i^{10}$

Step 15

Factor out $i^{4}$ .

$i^{4} i^{2} \cdot i^{10}$

Step 16

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

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Step 16.1

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

${(i^{2})}^{2} i^{2} \cdot i^{10}$

Step 16.2

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

${(- 1)}^{2} i^{2} \cdot i^{10}$

Step 16.3

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

$1 i^{2} \cdot i^{10}$

Step 17

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

$i^{2} \cdot i^{10}$

Step 18

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

$- 1 \cdot i^{10}$

Step 19

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

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Step 19.1

Factor out $i^{8}$ .

$- 1 \cdot (i^{8} i^{2})$

Step 19.2

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

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

Step 20

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

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Step 20.1

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

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

Step 20.2

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

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

Step 20.3

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

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

Step 21

One to any power is one.

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

Step 22

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

$- 1 \cdot i^{2}$

Step 23

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

$- 1 \cdot - 1$

Step 24

Multiply $- 1$ by $- 1$ .

$1$