Algebra Examples

$i^{13} + i^{18} + i^{31} + n = 0$

Step 1

Simplify $i^{13} + i^{18} + i^{31} + n$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $i^{13}$ as ${(i^{4})}^{3} i$ .

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Step 1.1.1.1

Factor out $i^{12}$ .

$i^{12} i + i^{18} + i^{31} + n = 0$

Step 1.1.1.2

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

${(i^{4})}^{3} i + i^{18} + i^{31} + n = 0$

Step 1.1.2

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

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Step 1.1.2.1

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

${({(i^{2})}^{2})}^{3} i + i^{18} + i^{31} + n = 0$

Step 1.1.2.2

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

${({(- 1)}^{2})}^{3} i + i^{18} + i^{31} + n = 0$

Step 1.1.2.3

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

$1^{3} i + i^{18} + i^{31} + n = 0$

Step 1.1.3

One to any power is one.

$1 i + i^{18} + i^{31} + n = 0$

Step 1.1.4

Multiply $i$ by $1$ .

$i + i^{18} + i^{31} + n = 0$

Step 1.1.5

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

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Step 1.1.5.1

Factor out $i^{16}$ .

$i + i^{16} i^{2} + i^{31} + n = 0$

Step 1.1.5.2

Rewrite $i^{16}$ as ${(i^{4})}^{4}$ .

$i + {(i^{4})}^{4} i^{2} + i^{31} + n = 0$

Step 1.1.6

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

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Step 1.1.6.1

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

$i + {({(i^{2})}^{2})}^{4} i^{2} + i^{31} + n = 0$

Step 1.1.6.2

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

$i + {({(- 1)}^{2})}^{4} i^{2} + i^{31} + n = 0$

Step 1.1.6.3

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

$i + 1^{4} i^{2} + i^{31} + n = 0$

Step 1.1.7

One to any power is one.

$i + 1 i^{2} + i^{31} + n = 0$

Step 1.1.8

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

$i + i^{2} + i^{31} + n = 0$

Step 1.1.9

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

$i - 1 + i^{31} + n = 0$

Step 1.1.10

Rewrite $i^{31}$ as ${(i^{4})}^{7} (i^{2} \cdot i)$ .

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Step 1.1.10.1

Factor out $i^{28}$ .

$i - 1 + i^{28} i^{3} + n = 0$

Step 1.1.10.2

Rewrite $i^{28}$ as ${(i^{4})}^{7}$ .

$i - 1 + {(i^{4})}^{7} i^{3} + n = 0$

Step 1.1.10.3

Factor out $i^{2}$ .

$i - 1 + {(i^{4})}^{7} (i^{2} \cdot i) + n = 0$

Step 1.1.11

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

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Step 1.1.11.1

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

$i - 1 + {({(i^{2})}^{2})}^{7} (i^{2} \cdot i) + n = 0$

Step 1.1.11.2

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

$i - 1 + {({(- 1)}^{2})}^{7} (i^{2} \cdot i) + n = 0$

Step 1.1.11.3

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

$i - 1 + 1^{7} (i^{2} \cdot i) + n = 0$

Step 1.1.12

One to any power is one.

$i - 1 + 1 (i^{2} \cdot i) + n = 0$

Step 1.1.13

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

$i - 1 + i^{2} \cdot i + n = 0$

Step 1.1.14

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

$i - 1 - 1 \cdot i + n = 0$

Step 1.1.15

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

$i - 1 - i + n = 0$

Step 1.2

Combine the opposite terms in $i - 1 - i + n$ .

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Step 1.2.1

Subtract $i$ from $i$ .

$0 - 1 + n = 0$

Step 1.2.2

Subtract $1$ from $0$ .

$- 1 + n = 0$

Step 2

Add $1$ to both sides of the equation.

$n = 1$