Precalculus Examples

$i^{105} - i^{- 222} + i^{- 32} - 2 i$

Step 1

Simplify each term.

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Step 1.1

Rewrite $i^{105}$ as ${(i^{4})}^{26} i$ .

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Step 1.1.1

Factor out $i^{104}$ .

$i^{104} i - i^{- 222} + i^{- 32} - 2 i$

Step 1.1.2

Rewrite $i^{104}$ as ${(i^{4})}^{26}$ .

${(i^{4})}^{26} i - i^{- 222} + i^{- 32} - 2 i$

Step 1.2

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

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Step 1.2.1

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

${({(i^{2})}^{2})}^{26} i - i^{- 222} + i^{- 32} - 2 i$

Step 1.2.2

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

${({(- 1)}^{2})}^{26} i - i^{- 222} + i^{- 32} - 2 i$

Step 1.2.3

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

$1^{26} i - i^{- 222} + i^{- 32} - 2 i$

Step 1.3

One to any power is one.

$1 i - i^{- 222} + i^{- 32} - 2 i$

Step 1.4

Multiply $i$ by $1$ .

$i - i^{- 222} + i^{- 32} - 2 i$

Step 1.5

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$i - \frac{1}{i^{222}} + i^{- 32} - 2 i$

Step 1.6

Simplify the denominator.

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Step 1.6.1

Rewrite $i^{222}$ as ${(i^{4})}^{55} i^{2}$ .

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Step 1.6.1.1

Factor out $i^{220}$ .

$i - \frac{1}{i^{220} i^{2}} + i^{- 32} - 2 i$

Step 1.6.1.2

Rewrite $i^{220}$ as ${(i^{4})}^{55}$ .

$i - \frac{1}{{(i^{4})}^{55} i^{2}} + i^{- 32} - 2 i$

Step 1.6.2

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

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Step 1.6.2.1

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

$i - \frac{1}{{({(i^{2})}^{2})}^{55} i^{2}} + i^{- 32} - 2 i$

Step 1.6.2.2

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

$i - \frac{1}{{({(- 1)}^{2})}^{55} i^{2}} + i^{- 32} - 2 i$

Step 1.6.2.3

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

$i - \frac{1}{1^{55} i^{2}} + i^{- 32} - 2 i$

Step 1.6.3

One to any power is one.

$i - \frac{1}{1 i^{2}} + i^{- 32} - 2 i$

Step 1.6.4

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

$i - \frac{1}{i^{2}} + i^{- 32} - 2 i$

Step 1.6.5

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

$i - \frac{1}{- 1} + i^{- 32} - 2 i$

Step 1.7

Divide $1$ by $- 1$ .

$i - - 1 + i^{- 32} - 2 i$

Step 1.8

Multiply $- 1$ by $- 1$ .

$i + 1 + i^{- 32} - 2 i$

Step 1.9

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$i + 1 + \frac{1}{i^{32}} - 2 i$

Step 1.10

Simplify the denominator.

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Step 1.10.1

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

$i + 1 + \frac{1}{{(i^{4})}^{8}} - 2 i$

Step 1.10.2

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

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Step 1.10.2.1

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

$i + 1 + \frac{1}{{({(i^{2})}^{2})}^{8}} - 2 i$

Step 1.10.2.2

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

$i + 1 + \frac{1}{{({(- 1)}^{2})}^{8}} - 2 i$

Step 1.10.2.3

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

$i + 1 + \frac{1}{1^{8}} - 2 i$

Step 1.10.3

One to any power is one.

$i + 1 + \frac{1}{1} - 2 i$

Step 1.11

Divide $1$ by $1$ .

$i + 1 + 1 - 2 i$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $2 i$ from $i$ .

$1 + 1 - i$

Step 2.2

Add $1$ and $1$ .

$2 - i$