Algebra Examples

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

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

Step 1

Rewrite

i^{34}

$i^{34}$ as

{(i^{4})}^{8} i^{2}

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

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Step 1.1

Factor out

i^{32}

$i^{32}$ .

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

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

Step 1.2

Rewrite

i^{32}

$i^{32}$ as

{(i^{4})}^{8}

${(i^{4})}^{8}$ .

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

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

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

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

Step 2

Rewrite

i^{4}

$i^{4}$ as

1

$1$ .

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Step 2.1

Rewrite

i^{4}

$i^{4}$ as

{(i^{2})}^{2}

${(i^{2})}^{2}$ .

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

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

Step 2.2

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

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

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

Step 2.3

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

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

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

Step 3

One to any power is one.

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

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

Step 4

Multiply

i^{2}

$i^{2}$ by

1

$1$ .

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

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

Step 5

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

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

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

Step 6

Rewrite

i^{15}

$i^{15}$ as

{(i^{4})}^{3} (i^{2} \cdot i)

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

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Step 6.1

Factor out

i^{12}

$i^{12}$ .

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

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

Step 6.2

Rewrite

i^{12}

$i^{12}$ as

{(i^{4})}^{3}

${(i^{4})}^{3}$ .

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

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

Step 6.3

Factor out

i^{2}

$i^{2}$ .

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

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

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

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

Step 7

Rewrite

i^{4}

$i^{4}$ as

1

$1$ .

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Step 7.1

Rewrite

i^{4}

$i^{4}$ as

{(i^{2})}^{2}

${(i^{2})}^{2}$ .

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

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

Step 7.2

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

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

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

Step 7.3

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

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

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

Step 8

One to any power is one.

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

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

Step 9

Multiply

i^{2} \cdot i

$i^{2} \cdot i$ by

1

$1$ .

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

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

Step 10

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

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

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

Step 11

Rewrite

- 1 i

$- 1 i$ as

- i

$- i$ .

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

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

Step 12

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

1 i \cdot i^{39}

$1 i \cdot i^{39}$

Step 13

Multiply

i

$i$ by

1

$1$ .

i \cdot i^{39}

$i \cdot i^{39}$

Step 14

Rewrite

i^{39}

$i^{39}$ as

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

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

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Step 14.1

Factor out

i^{36}

$i^{36}$ .

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

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

Step 14.2

Rewrite

i^{36}

$i^{36}$ as

{(i^{4})}^{9}

${(i^{4})}^{9}$ .

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

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

Step 14.3

Factor out

i^{2}

$i^{2}$ .

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

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

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

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

Step 15

Rewrite

i^{4}

$i^{4}$ as

1

$1$ .

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Step 15.1

Rewrite

i^{4}

$i^{4}$ as

{(i^{2})}^{2}

${(i^{2})}^{2}$ .

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

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

Step 15.2

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

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

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

Step 15.3

Raise

- 1

$- 1$ to the power of

2

$2$ .

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

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

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

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

Step 16

One to any power is one.

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

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

Step 17

Multiply

i^{2} \cdot i

$i^{2} \cdot i$ by

1

$1$ .

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

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

Step 18

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

i \cdot (- 1 \cdot i)

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

Step 19

Rewrite

- 1 i

$- 1 i$ as

- i

$- i$ .

i \cdot (- i)

$i \cdot (- i)$

Step 20

Multiply

i (- i)

$i (- i)$ .

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Step 20.1

Raise

i

$i$ to the power of

1

$1$ .

- (i^{1} i)

$- (i^{1} i)$

Step 20.2

Raise

i

$i$ to the power of

1

$1$ .

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

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

Step 20.3

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

- i^{1 + 1}

$- i^{1 + 1}$

Step 20.4

Add

1

$1$ and

1

$1$ .

- i^{2}

$- i^{2}$

- i^{2}

$- i^{2}$

Step 21

Rewrite

i^{2}

$i^{2}$ as

- 1

$- 1$ .

- - 1

$- - 1$

Step 22

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

1

$1$