Algebra Examples

$11 i^{59} + 6 i^{12} + 9 i^{15} + i^{11}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $i^{59}$ as ${(i^{4})}^{14} (i^{2} \cdot i)$ .

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Step 1.1.1

Factor out $i^{56}$ .

$11 (i^{56} i^{3}) + 6 i^{12} + 9 i^{15} + i^{11}$

Step 1.1.2

Rewrite $i^{56}$ as ${(i^{4})}^{14}$ .

$11 ({(i^{4})}^{14} i^{3}) + 6 i^{12} + 9 i^{15} + i^{11}$

Step 1.1.3

Factor out $i^{2}$ .

$11 ({(i^{4})}^{14} (i^{2} \cdot i)) + 6 i^{12} + 9 i^{15} + i^{11}$

Step 1.2

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

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Step 1.2.1

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

$11 ({({(i^{2})}^{2})}^{14} (i^{2} \cdot i)) + 6 i^{12} + 9 i^{15} + i^{11}$

Step 1.2.2

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

$11 ({({(- 1)}^{2})}^{14} (i^{2} \cdot i)) + 6 i^{12} + 9 i^{15} + i^{11}$

Step 1.2.3

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

$11 (1^{14} (i^{2} \cdot i)) + 6 i^{12} + 9 i^{15} + i^{11}$

Step 1.3

One to any power is one.

$11 (1 (i^{2} \cdot i)) + 6 i^{12} + 9 i^{15} + i^{11}$

Step 1.4

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

$11 (i^{2} \cdot i) + 6 i^{12} + 9 i^{15} + i^{11}$

Step 1.5

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

$11 (- 1 \cdot i) + 6 i^{12} + 9 i^{15} + i^{11}$

Step 1.6

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

$11 (- i) + 6 i^{12} + 9 i^{15} + i^{11}$

Step 1.7

Multiply $- 1$ by $11$ .

$- 11 i + 6 i^{12} + 9 i^{15} + i^{11}$

Step 1.8

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

$- 11 i + 6 {(i^{4})}^{3} + 9 i^{15} + i^{11}$

Step 1.9

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

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Step 1.9.1

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

$- 11 i + 6 {({(i^{2})}^{2})}^{3} + 9 i^{15} + i^{11}$

Step 1.9.2

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

$- 11 i + 6 {({(- 1)}^{2})}^{3} + 9 i^{15} + i^{11}$

Step 1.9.3

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

$- 11 i + 6 \cdot 1^{3} + 9 i^{15} + i^{11}$

Step 1.10

One to any power is one.

$- 11 i + 6 \cdot 1 + 9 i^{15} + i^{11}$

Step 1.11

Multiply $6$ by $1$ .

$- 11 i + 6 + 9 i^{15} + i^{11}$

Step 1.12

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

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Step 1.12.1

Factor out $i^{12}$ .

$- 11 i + 6 + 9 (i^{12} i^{3}) + i^{11}$

Step 1.12.2

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

$- 11 i + 6 + 9 ({(i^{4})}^{3} i^{3}) + i^{11}$

Step 1.12.3

Factor out $i^{2}$ .

$- 11 i + 6 + 9 ({(i^{4})}^{3} (i^{2} \cdot i)) + i^{11}$

Step 1.13

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

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Step 1.13.1

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

$- 11 i + 6 + 9 ({({(i^{2})}^{2})}^{3} (i^{2} \cdot i)) + i^{11}$

Step 1.13.2

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

$- 11 i + 6 + 9 ({({(- 1)}^{2})}^{3} (i^{2} \cdot i)) + i^{11}$

Step 1.13.3

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

$- 11 i + 6 + 9 (1^{3} (i^{2} \cdot i)) + i^{11}$

Step 1.14

One to any power is one.

$- 11 i + 6 + 9 (1 (i^{2} \cdot i)) + i^{11}$

Step 1.15

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

$- 11 i + 6 + 9 (i^{2} \cdot i) + i^{11}$

Step 1.16

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

$- 11 i + 6 + 9 (- 1 \cdot i) + i^{11}$

Step 1.17

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

$- 11 i + 6 + 9 (- i) + i^{11}$

Step 1.18

Multiply $- 1$ by $9$ .

$- 11 i + 6 - 9 i + i^{11}$

Step 1.19

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

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Step 1.19.1

Factor out $i^{8}$ .

$- 11 i + 6 - 9 i + i^{8} i^{3}$

Step 1.19.2

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

$- 11 i + 6 - 9 i + {(i^{4})}^{2} i^{3}$

Step 1.19.3

Factor out $i^{2}$ .

$- 11 i + 6 - 9 i + {(i^{4})}^{2} (i^{2} \cdot i)$

Step 1.20

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

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Step 1.20.1

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

$- 11 i + 6 - 9 i + {({(i^{2})}^{2})}^{2} (i^{2} \cdot i)$

Step 1.20.2

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

$- 11 i + 6 - 9 i + {({(- 1)}^{2})}^{2} (i^{2} \cdot i)$

Step 1.20.3

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

$- 11 i + 6 - 9 i + 1^{2} (i^{2} \cdot i)$

Step 1.21

One to any power is one.

$- 11 i + 6 - 9 i + 1 (i^{2} \cdot i)$

Step 1.22

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

$- 11 i + 6 - 9 i + i^{2} \cdot i$

Step 1.23

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

$- 11 i + 6 - 9 i - 1 \cdot i$

Step 1.24

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

$- 11 i + 6 - 9 i - i$

Step 2

Simplify by adding terms.

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Step 2.1

Subtract $9 i$ from $- 11 i$ .

$6 - 20 i - i$

Step 2.2

Subtract $i$ from $- 20 i$ .

$6 - 21 i$