Algebra Examples

$i^{6} \cdot i^{19} \cdot i^{39}$

Step 1

Factor out $i^{4}$ .

$i^{4} i^{2} \cdot i^{19} \cdot i^{39}$

Step 2

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

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Step 2.1

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

${(i^{2})}^{2} i^{2} \cdot i^{19} \cdot i^{39}$

Step 2.2

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

${(- 1)}^{2} i^{2} \cdot i^{19} \cdot i^{39}$

Step 2.3

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

$1 i^{2} \cdot i^{19} \cdot i^{39}$

Step 3

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

$i^{2} \cdot i^{19} \cdot i^{39}$

Step 4

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

$- 1 \cdot i^{19} \cdot i^{39}$

Step 5

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

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Step 5.1

Factor out $i^{16}$ .

$- 1 \cdot (i^{16} i^{3}) \cdot i^{39}$

Step 5.2

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

$- 1 \cdot ({(i^{4})}^{4} i^{3}) \cdot i^{39}$

Step 5.3

Factor out $i^{2}$ .

$- 1 \cdot ({(i^{4})}^{4} (i^{2} \cdot i)) \cdot i^{39}$

Step 6

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

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Step 6.1

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

$- 1 \cdot ({({(i^{2})}^{2})}^{4} (i^{2} \cdot i)) \cdot i^{39}$

Step 6.2

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

$- 1 \cdot ({({(- 1)}^{2})}^{4} (i^{2} \cdot i)) \cdot i^{39}$

Step 6.3

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

$- 1 \cdot (1^{4} (i^{2} \cdot i)) \cdot i^{39}$

Step 7

One to any power is one.

$- 1 \cdot (1 (i^{2} \cdot i)) \cdot i^{39}$

Step 8

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

$- 1 \cdot (i^{2} \cdot i) \cdot i^{39}$

Step 9

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

$- 1 \cdot (- 1 \cdot i) \cdot i^{39}$

Step 10

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

$- 1 \cdot (- i) \cdot i^{39}$

Step 11

Multiply $- 1$ by $- 1$ .

$1 i \cdot i^{39}$

Step 12

Multiply $i$ by $1$ .

$i \cdot i^{39}$

Step 13

Rewrite $i^{39}$ as ${(i^{4})}^{9} (i^{2} \cdot i)$ .

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Step 13.1

Factor out $i^{36}$ .

$i \cdot (i^{36} i^{3})$

Step 13.2

Rewrite $i^{36}$ as ${(i^{4})}^{9}$ .

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

Step 13.3

Factor out $i^{2}$ .

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

Step 14

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

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Step 14.1

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

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

Step 14.2

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

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

Step 14.3

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

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

Step 15

One to any power is one.

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

Step 16

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

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

Step 17

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

$i \cdot (- 1 \cdot i)$

Step 18

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

$i \cdot (- i)$

Step 19

Multiply $i (- i)$ .

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Step 19.1

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

$- (i^{1} i)$

Step 19.2

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

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

Step 19.3

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

$- i^{1 + 1}$

Step 19.4

Add $1$ and $1$ .

$- i^{2}$

Step 20

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

$- - 1$

Step 21

Multiply $- 1$ by $- 1$ .

$1$