Algebra Examples

$i^{19} \cdot i^{16} \cdot i^{15} \cdot i^{17}$

Step 1

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

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Step 1.1

Factor out $i^{16}$ .

$i^{16} i^{3} \cdot i^{16} \cdot i^{15} \cdot i^{17}$

Step 1.2

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

${(i^{4})}^{4} i^{3} \cdot i^{16} \cdot i^{15} \cdot i^{17}$

Step 1.3

Factor out $i^{2}$ .

${(i^{4})}^{4} (i^{2} \cdot i) \cdot i^{16} \cdot i^{15} \cdot i^{17}$

Step 2

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

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Step 2.1

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

${({(i^{2})}^{2})}^{4} (i^{2} \cdot i) \cdot i^{16} \cdot i^{15} \cdot i^{17}$

Step 2.2

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

${({(- 1)}^{2})}^{4} (i^{2} \cdot i) \cdot i^{16} \cdot i^{15} \cdot i^{17}$

Step 2.3

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

$1^{4} (i^{2} \cdot i) \cdot i^{16} \cdot i^{15} \cdot i^{17}$

Step 3

One to any power is one.

$1 (i^{2} \cdot i) \cdot i^{16} \cdot i^{15} \cdot i^{17}$

Step 4

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

$i^{2} \cdot i \cdot i^{16} \cdot i^{15} \cdot i^{17}$

Step 5

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

$- 1 \cdot i \cdot i^{16} \cdot i^{15} \cdot i^{17}$

Step 6

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

$- i \cdot i^{16} \cdot i^{15} \cdot i^{17}$

Step 7

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

$- i \cdot {(i^{4})}^{4} \cdot i^{15} \cdot i^{17}$

Step 8

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

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Step 8.1

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

$- i \cdot {({(i^{2})}^{2})}^{4} \cdot i^{15} \cdot i^{17}$

Step 8.2

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

$- i \cdot {({(- 1)}^{2})}^{4} \cdot i^{15} \cdot i^{17}$

Step 8.3

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

$- i \cdot 1^{4} \cdot i^{15} \cdot i^{17}$

Step 9

One to any power is one.

$- i \cdot 1 \cdot i^{15} \cdot i^{17}$

Step 10

Multiply $- 1$ by $1$ .

$- i \cdot i^{15} \cdot i^{17}$

Step 11

Rewrite $i^{15}$ as ${(i^{4})}^{3} (i^{2} \cdot i)$ .

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Step 11.1

Factor out $i^{12}$ .

$- i \cdot (i^{12} i^{3}) \cdot i^{17}$

Step 11.2

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

$- i \cdot ({(i^{4})}^{3} i^{3}) \cdot i^{17}$

Step 11.3

Factor out $i^{2}$ .

$- i \cdot ({(i^{4})}^{3} (i^{2} \cdot i)) \cdot i^{17}$

Step 12

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

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Step 12.1

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

$- i \cdot ({({(i^{2})}^{2})}^{3} (i^{2} \cdot i)) \cdot i^{17}$

Step 12.2

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

$- i \cdot ({({(- 1)}^{2})}^{3} (i^{2} \cdot i)) \cdot i^{17}$

Step 12.3

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

$- i \cdot (1^{3} (i^{2} \cdot i)) \cdot i^{17}$

Step 13

One to any power is one.

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

Step 14

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

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

Step 15

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

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

Step 16

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

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

Step 17

Multiply $- i (- i)$ .

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Step 17.1

Multiply $- 1$ by $- 1$ .

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

Step 17.2

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

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

Step 17.3

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

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

Step 17.4

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

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

Step 17.5

Add $1$ and $1$ .

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

Step 17.6

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

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

Step 18

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

$- 1 \cdot i^{17}$

Step 19

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

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Step 19.1

Factor out $i^{16}$ .

$- 1 \cdot (i^{16} i)$

Step 19.2

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

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

Step 20

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

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Step 20.1

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

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

Step 20.2

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

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

Step 20.3

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

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

Step 21

One to any power is one.

$- 1 \cdot (1 i)$

Step 22

Multiply $i$ by $1$ .

$- 1 \cdot i$

Step 23

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

$- i$