Linear Algebra Examples

$i^{22} + i^{33} + i^{44} + i^{55}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $i^{22}$ as ${(i^{4})}^{5} i^{2}$ .

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Step 1.1.1

Factor out $i^{20}$ .

$i^{20} i^{2} + i^{33} + i^{44} + i^{55}$

Step 1.1.2

Rewrite $i^{20}$ as ${(i^{4})}^{5}$ .

${(i^{4})}^{5} i^{2} + i^{33} + i^{44} + i^{55}$

${(i^{4})}^{5} i^{2} + i^{33} + i^{44} + i^{55}$

Step 1.2

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

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Step 1.2.1

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

${({(i^{2})}^{2})}^{5} i^{2} + i^{33} + i^{44} + i^{55}$

Step 1.2.2

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

${({(- 1)}^{2})}^{5} i^{2} + i^{33} + i^{44} + i^{55}$

Step 1.2.3

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

$1^{5} i^{2} + i^{33} + i^{44} + i^{55}$

$1^{5} i^{2} + i^{33} + i^{44} + i^{55}$

Step 1.3

One to any power is one.

$1 i^{2} + i^{33} + i^{44} + i^{55}$

Step 1.4

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

$i^{2} + i^{33} + i^{44} + i^{55}$

Step 1.5

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

$- 1 + i^{33} + i^{44} + i^{55}$

Step 1.6

Rewrite $i^{33}$ as ${(i^{4})}^{8} i$ .

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Step 1.6.1

Factor out $i^{32}$ .

$- 1 + i^{32} i + i^{44} + i^{55}$

Step 1.6.2

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

$- 1 + {(i^{4})}^{8} i + i^{44} + i^{55}$

$- 1 + {(i^{4})}^{8} i + i^{44} + i^{55}$

Step 1.7

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

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Step 1.7.1

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

$- 1 + {({(i^{2})}^{2})}^{8} i + i^{44} + i^{55}$

Step 1.7.2

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

$- 1 + {({(- 1)}^{2})}^{8} i + i^{44} + i^{55}$

Step 1.7.3

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

$- 1 + 1^{8} i + i^{44} + i^{55}$

$- 1 + 1^{8} i + i^{44} + i^{55}$

Step 1.8

One to any power is one.

$- 1 + 1 i + i^{44} + i^{55}$

Step 1.9

Multiply $i$ by $1$ .

$- 1 + i + i^{44} + i^{55}$

Step 1.10

Rewrite $i^{44}$ as ${(i^{4})}^{11}$ .

$- 1 + i + {(i^{4})}^{11} + i^{55}$

Step 1.11

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

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Step 1.11.1

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

$- 1 + i + {({(i^{2})}^{2})}^{11} + i^{55}$

Step 1.11.2

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

$- 1 + i + {({(- 1)}^{2})}^{11} + i^{55}$

Step 1.11.3

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

$- 1 + i + 1^{11} + i^{55}$

$- 1 + i + 1^{11} + i^{55}$

Step 1.12

One to any power is one.

$- 1 + i + 1 + i^{55}$

Step 1.13

Rewrite $i^{55}$ as ${(i^{4})}^{13} (i^{2} \cdot i)$ .

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Step 1.13.1

Factor out $i^{52}$ .

$- 1 + i + 1 + i^{52} i^{3}$

Step 1.13.2

Rewrite $i^{52}$ as ${(i^{4})}^{13}$ .

$- 1 + i + 1 + {(i^{4})}^{13} i^{3}$

Step 1.13.3

Factor out $i^{2}$ .

$- 1 + i + 1 + {(i^{4})}^{13} (i^{2} \cdot i)$

$- 1 + i + 1 + {(i^{4})}^{13} (i^{2} \cdot i)$

Step 1.14

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

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Step 1.14.1

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

$- 1 + i + 1 + {({(i^{2})}^{2})}^{13} (i^{2} \cdot i)$

Step 1.14.2

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

$- 1 + i + 1 + {({(- 1)}^{2})}^{13} (i^{2} \cdot i)$

Step 1.14.3

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

$- 1 + i + 1 + 1^{13} (i^{2} \cdot i)$

$- 1 + i + 1 + 1^{13} (i^{2} \cdot i)$

Step 1.15

One to any power is one.

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

Step 1.16

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

$- 1 + i + 1 + i^{2} \cdot i$

Step 1.17

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

$- 1 + i + 1 - 1 \cdot i$

Step 1.18

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

$- 1 + i + 1 - i$

$- 1 + i + 1 - i$

Step 2

Simplify by adding terms.

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Step 2.1

Add $- 1$ and $1$ .

$0 + i - i$

Step 2.2

Add $0$ and $i$ .

$i - i$

Step 2.3

Subtract $i$ from $i$ .

$0$

$0$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$