Algebra Examples

$i^{119} + i^{42} + 5 i^{57} - 8 i^{74}$

Step 1

Simplify each term.

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Step 1.1

Rewrite $i^{119}$ as ${(i^{4})}^{29} (i^{2} \cdot i)$ .

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Step 1.1.1

Factor out $i^{116}$ .

$i^{116} i^{3} + i^{42} + 5 i^{57} - 8 i^{74}$

Step 1.1.2

Rewrite $i^{116}$ as ${(i^{4})}^{29}$ .

${(i^{4})}^{29} i^{3} + i^{42} + 5 i^{57} - 8 i^{74}$

Step 1.1.3

Factor out $i^{2}$ .

${(i^{4})}^{29} (i^{2} \cdot i) + i^{42} + 5 i^{57} - 8 i^{74}$

Step 1.2

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

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Step 1.2.1

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

${({(i^{2})}^{2})}^{29} (i^{2} \cdot i) + i^{42} + 5 i^{57} - 8 i^{74}$

Step 1.2.2

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

${({(- 1)}^{2})}^{29} (i^{2} \cdot i) + i^{42} + 5 i^{57} - 8 i^{74}$

Step 1.2.3

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

$1^{29} (i^{2} \cdot i) + i^{42} + 5 i^{57} - 8 i^{74}$

Step 1.3

One to any power is one.

$1 (i^{2} \cdot i) + i^{42} + 5 i^{57} - 8 i^{74}$

Step 1.4

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

$i^{2} \cdot i + i^{42} + 5 i^{57} - 8 i^{74}$

Step 1.5

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

$- 1 \cdot i + i^{42} + 5 i^{57} - 8 i^{74}$

Step 1.6

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

$- i + i^{42} + 5 i^{57} - 8 i^{74}$

Step 1.7

Rewrite $i^{42}$ as ${(i^{4})}^{10} i^{2}$ .

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Step 1.7.1

Factor out $i^{40}$ .

$- i + i^{40} i^{2} + 5 i^{57} - 8 i^{74}$

Step 1.7.2

Rewrite $i^{40}$ as ${(i^{4})}^{10}$ .

$- i + {(i^{4})}^{10} i^{2} + 5 i^{57} - 8 i^{74}$

Step 1.8

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

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Step 1.8.1

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

$- i + {({(i^{2})}^{2})}^{10} i^{2} + 5 i^{57} - 8 i^{74}$

Step 1.8.2

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

$- i + {({(- 1)}^{2})}^{10} i^{2} + 5 i^{57} - 8 i^{74}$

Step 1.8.3

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

$- i + 1^{10} i^{2} + 5 i^{57} - 8 i^{74}$

Step 1.9

One to any power is one.

$- i + 1 i^{2} + 5 i^{57} - 8 i^{74}$

Step 1.10

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

$- i + i^{2} + 5 i^{57} - 8 i^{74}$

Step 1.11

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

$- i - 1 + 5 i^{57} - 8 i^{74}$

Step 1.12

Rewrite $i^{57}$ as ${(i^{4})}^{14} i$ .

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Step 1.12.1

Factor out $i^{56}$ .

$- i - 1 + 5 (i^{56} i) - 8 i^{74}$

Step 1.12.2

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

$- i - 1 + 5 ({(i^{4})}^{14} i) - 8 i^{74}$

Step 1.13

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

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Step 1.13.1

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

$- i - 1 + 5 ({({(i^{2})}^{2})}^{14} i) - 8 i^{74}$

Step 1.13.2

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

$- i - 1 + 5 ({({(- 1)}^{2})}^{14} i) - 8 i^{74}$

Step 1.13.3

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

$- i - 1 + 5 (1^{14} i) - 8 i^{74}$

Step 1.14

One to any power is one.

$- i - 1 + 5 (1 i) - 8 i^{74}$

Step 1.15

Multiply $i$ by $1$ .

$- i - 1 + 5 i - 8 i^{74}$

Step 1.16

Rewrite $i^{74}$ as ${(i^{4})}^{18} i^{2}$ .

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Step 1.16.1

Factor out $i^{72}$ .

$- i - 1 + 5 i - 8 (i^{72} i^{2})$

Step 1.16.2

Rewrite $i^{72}$ as ${(i^{4})}^{18}$ .

$- i - 1 + 5 i - 8 ({(i^{4})}^{18} i^{2})$

Step 1.17

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

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Step 1.17.1

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

$- i - 1 + 5 i - 8 ({({(i^{2})}^{2})}^{18} i^{2})$

Step 1.17.2

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

$- i - 1 + 5 i - 8 ({({(- 1)}^{2})}^{18} i^{2})$

Step 1.17.3

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

$- i - 1 + 5 i - 8 (1^{18} i^{2})$

Step 1.18

One to any power is one.

$- i - 1 + 5 i - 8 (1 i^{2})$

Step 1.19

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

$- i - 1 + 5 i - 8 i^{2}$

Step 1.20

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

$- i - 1 + 5 i - 8 \cdot - 1$

Step 1.21

Multiply $- 8$ by $- 1$ .

$- i - 1 + 5 i + 8$

Step 2

Simplify by adding terms.

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Step 2.1

Add $- i$ and $5 i$ .

$- 1 + 4 i + 8$

Step 2.2

Add $- 1$ and $8$ .

$7 + 4 i$