Algebra Examples

$i^{34} \cdot i^{15} \cdot i^{39}$

Step 1

Rewrite $i^{34}$ as ${(i^{4})}^{8} i^{2}$ .

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Step 1.1

Factor out $i^{32}$ .

$i^{32} i^{2} \cdot i^{15} \cdot i^{39}$

Step 1.2

Rewrite $i^{32}$ as ${(i^{4})}^{8}$ .

${(i^{4})}^{8} i^{2} \cdot i^{15} \cdot i^{39}$

Step 2

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

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Step 2.1

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

${({(i^{2})}^{2})}^{8} i^{2} \cdot i^{15} \cdot i^{39}$

Step 2.2

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

${({(- 1)}^{2})}^{8} i^{2} \cdot i^{15} \cdot i^{39}$

Step 2.3

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

$1^{8} i^{2} \cdot i^{15} \cdot i^{39}$

Step 3

One to any power is one.

$1 i^{2} \cdot i^{15} \cdot i^{39}$

Step 4

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

$i^{2} \cdot i^{15} \cdot i^{39}$

Step 5

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

$- 1 \cdot i^{15} \cdot i^{39}$

Step 6

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

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Step 6.1

Factor out $i^{12}$ .

$- 1 \cdot (i^{12} i^{3}) \cdot i^{39}$

Step 6.2

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

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

Step 6.3

Factor out $i^{2}$ .

$- 1 \cdot ({(i^{4})}^{3} (i^{2} \cdot i)) \cdot i^{39}$

Step 7

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

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Step 7.1

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

$- 1 \cdot ({({(i^{2})}^{2})}^{3} (i^{2} \cdot i)) \cdot i^{39}$

Step 7.2

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

$- 1 \cdot ({({(- 1)}^{2})}^{3} (i^{2} \cdot i)) \cdot i^{39}$

Step 7.3

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

$- 1 \cdot (1^{3} (i^{2} \cdot i)) \cdot i^{39}$

Step 8

One to any power is one.

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

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Multiply $- 1$ by $- 1$ .

$1 i \cdot i^{39}$

Step 13

Multiply $i$ by $1$ .

$i \cdot i^{39}$

Step 14

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

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Step 14.1

Factor out $i^{36}$ .

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

Step 14.2

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

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

Step 14.3

Factor out $i^{2}$ .

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

Step 15

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

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Step 15.1

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

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

Step 15.2

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

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

Step 15.3

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

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

Step 16

One to any power is one.

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

Step 17

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

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

Step 18

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

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

Step 19

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

$i \cdot (- i)$

Step 20

Multiply $i (- i)$ .

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Step 20.1

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

$- (i^{1} i)$

Step 20.2

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

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

Step 20.3

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

$- i^{1 + 1}$

Step 20.4

Add $1$ and $1$ .

$- i^{2}$

Step 21

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

$- - 1$

Step 22

Multiply $- 1$ by $- 1$ .

$1$