Algebra Examples

$i^{19} \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Step 1

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

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Step 1.1

Factor out $i^{16}$ .

$i^{16} i^{3} \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Step 1.2

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

${(i^{4})}^{4} i^{3} \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Step 1.3

Factor out $i^{2}$ .

${(i^{4})}^{4} (i^{2} \cdot i) \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Step 2

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

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Step 2.1

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

${({(i^{2})}^{2})}^{4} (i^{2} \cdot i) \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Step 2.2

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

${({(- 1)}^{2})}^{4} (i^{2} \cdot i) \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Step 2.3

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

$1^{4} (i^{2} \cdot i) \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Step 3

One to any power is one.

$1 (i^{2} \cdot i) \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Step 4

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

$i^{2} \cdot i \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Step 5

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

$- 1 \cdot i \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Step 6

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

$- i \cdot i^{23} \cdot i^{11} \cdot i^{21}$

Step 7

Rewrite $i^{23}$ as ${(i^{4})}^{5} (i^{2} \cdot i)$ .

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Step 7.1

Factor out $i^{20}$ .

$- i \cdot (i^{20} i^{3}) \cdot i^{11} \cdot i^{21}$

Step 7.2

Rewrite $i^{20}$ as ${(i^{4})}^{5}$ .

$- i \cdot ({(i^{4})}^{5} i^{3}) \cdot i^{11} \cdot i^{21}$

Step 7.3

Factor out $i^{2}$ .

$- i \cdot ({(i^{4})}^{5} (i^{2} \cdot i)) \cdot i^{11} \cdot i^{21}$

Step 8

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

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Step 8.1

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

$- i \cdot ({({(i^{2})}^{2})}^{5} (i^{2} \cdot i)) \cdot i^{11} \cdot i^{21}$

Step 8.2

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

$- i \cdot ({({(- 1)}^{2})}^{5} (i^{2} \cdot i)) \cdot i^{11} \cdot i^{21}$

Step 8.3

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

$- i \cdot (1^{5} (i^{2} \cdot i)) \cdot i^{11} \cdot i^{21}$

Step 9

One to any power is one.

$- i \cdot (1 (i^{2} \cdot i)) \cdot i^{11} \cdot i^{21}$

Step 10

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

$- i \cdot (i^{2} \cdot i) \cdot i^{11} \cdot i^{21}$

Step 11

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

$- i \cdot (- 1 \cdot i) \cdot i^{11} \cdot i^{21}$

Step 12

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

$- i \cdot (- i) \cdot i^{11} \cdot i^{21}$

Step 13

Multiply $- i (- i)$ .

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Step 13.1

Multiply $- 1$ by $- 1$ .

$1 i i \cdot i^{11} \cdot i^{21}$

Step 13.2

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

$1 (i^{1} i) \cdot i^{11} \cdot i^{21}$

Step 13.3

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

$1 (i^{1} i^{1}) \cdot i^{11} \cdot i^{21}$

Step 13.4

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

$1 i^{1 + 1} \cdot i^{11} \cdot i^{21}$

Step 13.5

Add $1$ and $1$ .

$1 i^{2} \cdot i^{11} \cdot i^{21}$

Step 13.6

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

$i^{2} \cdot i^{11} \cdot i^{21}$

Step 14

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

$- 1 \cdot i^{11} \cdot i^{21}$

Step 15

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

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Step 15.1

Factor out $i^{8}$ .

$- 1 \cdot (i^{8} i^{3}) \cdot i^{21}$

Step 15.2

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

$- 1 \cdot ({(i^{4})}^{2} i^{3}) \cdot i^{21}$

Step 15.3

Factor out $i^{2}$ .

$- 1 \cdot ({(i^{4})}^{2} (i^{2} \cdot i)) \cdot i^{21}$

Step 16

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

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Step 16.1

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

$- 1 \cdot ({({(i^{2})}^{2})}^{2} (i^{2} \cdot i)) \cdot i^{21}$

Step 16.2

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

$- 1 \cdot ({({(- 1)}^{2})}^{2} (i^{2} \cdot i)) \cdot i^{21}$

Step 16.3

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

$- 1 \cdot (1^{2} (i^{2} \cdot i)) \cdot i^{21}$

Step 17

One to any power is one.

$- 1 \cdot (1 (i^{2} \cdot i)) \cdot i^{21}$

Step 18

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

$- 1 \cdot (i^{2} \cdot i) \cdot i^{21}$

Step 19

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

$- 1 \cdot (- 1 \cdot i) \cdot i^{21}$

Step 20

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

$- 1 \cdot (- i) \cdot i^{21}$

Step 21

Multiply $- 1$ by $- 1$ .

$1 i \cdot i^{21}$

Step 22

Multiply $i$ by $1$ .

$i \cdot i^{21}$

Step 23

Rewrite $i^{21}$ as ${(i^{4})}^{5} i$ .

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Step 23.1

Factor out $i^{20}$ .

$i \cdot (i^{20} i)$

Step 23.2

Rewrite $i^{20}$ as ${(i^{4})}^{5}$ .

$i \cdot ({(i^{4})}^{5} i)$

Step 24

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

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Step 24.1

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

$i \cdot ({({(i^{2})}^{2})}^{5} i)$

Step 24.2

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

$i \cdot ({({(- 1)}^{2})}^{5} i)$

Step 24.3

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

$i \cdot (1^{5} i)$

Step 25

One to any power is one.

$i \cdot (1 i)$

Step 26

Multiply $i$ by $1$ .

$i \cdot i$

Step 27

Multiply $i i$ .

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Step 27.1

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

$i^{1} i$

Step 27.2

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

$i^{1} i^{1}$

Step 27.3

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

$i^{1 + 1}$

Step 27.4

Add $1$ and $1$ .

$i^{2}$

Step 28

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

$- 1$