Algebra Examples

$i^{11} \cdot i^{25} + i^{2021}$

Step 1

Rewrite $i^{11}$ as ${(i^{4})}^{2} (i^{2} \cdot i)$ .

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Step 1.1

Factor out $i^{8}$ .

$i^{8} i^{3} \cdot i^{25} + i^{2021}$

Step 1.2

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

${(i^{4})}^{2} i^{3} \cdot i^{25} + i^{2021}$

Step 1.3

Factor out $i^{2}$ .

${(i^{4})}^{2} (i^{2} \cdot i) \cdot i^{25} + i^{2021}$

Step 2

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

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Step 2.1

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

${({(i^{2})}^{2})}^{2} (i^{2} \cdot i) \cdot i^{25} + i^{2021}$

Step 2.2

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

${({(- 1)}^{2})}^{2} (i^{2} \cdot i) \cdot i^{25} + i^{2021}$

Step 2.3

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

$1^{2} (i^{2} \cdot i) \cdot i^{25} + i^{2021}$

Step 3

One to any power is one.

$1 (i^{2} \cdot i) \cdot i^{25} + i^{2021}$

Step 4

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

$i^{2} \cdot i \cdot i^{25} + i^{2021}$

Step 5

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

$- 1 \cdot i \cdot i^{25} + i^{2021}$

Step 6

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

$- i \cdot i^{25} + i^{2021}$

Step 7

Rewrite $i^{25}$ as ${(i^{4})}^{6} i$ .

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Step 7.1

Factor out $i^{24}$ .

$- i \cdot (i^{24} i) + i^{2021}$

Step 7.2

Rewrite $i^{24}$ as ${(i^{4})}^{6}$ .

$- i \cdot ({(i^{4})}^{6} i) + i^{2021}$

Step 8

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

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Step 8.1

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

$- i \cdot ({({(i^{2})}^{2})}^{6} i) + i^{2021}$

Step 8.2

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

$- i \cdot ({({(- 1)}^{2})}^{6} i) + i^{2021}$

Step 8.3

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

$- i \cdot (1^{6} i) + i^{2021}$

Step 9

One to any power is one.

$- i \cdot (1 i) + i^{2021}$

Step 10

Multiply $i$ by $1$ .

$- i \cdot i + i^{2021}$

Step 11

Multiply $- i i$ .

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Step 11.1

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

$- (i^{1} i) + i^{2021}$

Step 11.2

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

$- (i^{1} i^{1}) + i^{2021}$

Step 11.3

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

$- i^{1 + 1} + i^{2021}$

Step 11.4

Add $1$ and $1$ .

$- i^{2} + i^{2021}$

Step 12

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

$- - 1 + i^{2021}$

Step 13

Multiply $- 1$ by $- 1$ .

$1 + i^{2021}$

Step 14

Rewrite $i^{2021}$ as ${(i^{4})}^{505} i$ .

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Step 14.1

Factor out $i^{2020}$ .

$1 + i^{2020} i$

Step 14.2

Rewrite $i^{2020}$ as ${(i^{4})}^{505}$ .

$1 + {(i^{4})}^{505} i$

Step 15

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

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Step 15.1

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

$1 + {({(i^{2})}^{2})}^{505} i$

Step 15.2

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

$1 + {({(- 1)}^{2})}^{505} i$

Step 15.3

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

$1 + 1^{505} i$

Step 16

One to any power is one.

$1 + 1 i$

Step 17

Multiply $i$ by $1$ .

$1 + i$