Algebra Examples

$i^{11} \cdot i^{13} \cdot i^{7} \cdot i^{9}$

Step 1

Rewrite $i^{11}$ as ${(i^{4})}^{2} (i^{2} \cdot i)$ .

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Step 1.1

Factor out $i^{8}$ .

$i^{8} i^{3} \cdot i^{13} \cdot i^{7} \cdot i^{9}$

Step 1.2

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

${(i^{4})}^{2} i^{3} \cdot i^{13} \cdot i^{7} \cdot i^{9}$

Step 1.3

Factor out $i^{2}$ .

${(i^{4})}^{2} (i^{2} \cdot i) \cdot i^{13} \cdot i^{7} \cdot i^{9}$

Step 2

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

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Step 2.1

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

${({(i^{2})}^{2})}^{2} (i^{2} \cdot i) \cdot i^{13} \cdot i^{7} \cdot i^{9}$

Step 2.2

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

${({(- 1)}^{2})}^{2} (i^{2} \cdot i) \cdot i^{13} \cdot i^{7} \cdot i^{9}$

Step 2.3

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

$1^{2} (i^{2} \cdot i) \cdot i^{13} \cdot i^{7} \cdot i^{9}$

Step 3

One to any power is one.

$1 (i^{2} \cdot i) \cdot i^{13} \cdot i^{7} \cdot i^{9}$

Step 4

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

$i^{2} \cdot i \cdot i^{13} \cdot i^{7} \cdot i^{9}$

Step 5

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

$- 1 \cdot i \cdot i^{13} \cdot i^{7} \cdot i^{9}$

Step 6

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

$- i \cdot i^{13} \cdot i^{7} \cdot i^{9}$

Step 7

Rewrite $i^{13}$ as ${(i^{4})}^{3} i$ .

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Step 7.1

Factor out $i^{12}$ .

$- i \cdot (i^{12} i) \cdot i^{7} \cdot i^{9}$

Step 7.2

Rewrite $i^{12}$ as ${(i^{4})}^{3}$ .

$- i \cdot ({(i^{4})}^{3} i) \cdot i^{7} \cdot i^{9}$

Step 8

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

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Step 8.1

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

$- i \cdot ({({(i^{2})}^{2})}^{3} i) \cdot i^{7} \cdot i^{9}$

Step 8.2

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

$- i \cdot ({({(- 1)}^{2})}^{3} i) \cdot i^{7} \cdot i^{9}$

Step 8.3

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

$- i \cdot (1^{3} i) \cdot i^{7} \cdot i^{9}$

Step 9

One to any power is one.

$- i \cdot (1 i) \cdot i^{7} \cdot i^{9}$

Step 10

Multiply $i$ by $1$ .

$- i \cdot i \cdot i^{7} \cdot i^{9}$

Step 11

Multiply $- i i$ .

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Step 11.1

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

$- (i^{1} i) \cdot i^{7} \cdot i^{9}$

Step 11.2

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

$- (i^{1} i^{1}) \cdot i^{7} \cdot i^{9}$

Step 11.3

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

$- i^{1 + 1} \cdot i^{7} \cdot i^{9}$

Step 11.4

Add $1$ and $1$ .

$- i^{2} \cdot i^{7} \cdot i^{9}$

Step 12

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

$- - 1 \cdot i^{7} \cdot i^{9}$

Step 13

Multiply $- 1$ by $- 1$ .

$1 \cdot i^{7} \cdot i^{9}$

Step 14

Multiply $i^{7}$ by $1$ .

$i^{7} \cdot i^{9}$

Step 15

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

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Step 15.1

Factor out $i^{4}$ .

$i^{4} i^{3} \cdot i^{9}$

Step 15.2

Factor out $i^{2}$ .

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

Step 16

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

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Step 16.1

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

${(i^{2})}^{2} (i^{2} \cdot i) \cdot i^{9}$

Step 16.2

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

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

Step 16.3

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

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

Step 17

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

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

Step 18

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

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

Step 19

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

$- i \cdot i^{9}$

Step 20

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

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Step 20.1

Factor out $i^{8}$ .

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

Step 20.2

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

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

Step 21

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

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Step 21.1

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

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

Step 21.2

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

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

Step 21.3

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

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

Step 22

One to any power is one.

$- i \cdot (1 i)$

Step 23

Multiply $i$ by $1$ .

$- i \cdot i$

Step 24

Multiply $- i i$ .

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Step 24.1

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

$- (i^{1} i)$

Step 24.2

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

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

Step 24.3

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

$- i^{1 + 1}$

Step 24.4

Add $1$ and $1$ .

$- i^{2}$

Step 25

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

$- - 1$

Step 26

Multiply $- 1$ by $- 1$ .

$1$