Solve for n i^13+i^18+i^31+n=0

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

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

Simplify each term.

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

Factor out $i^{12}$ .

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

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

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

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

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

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

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

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

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

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

One to any power is one.

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

Multiply $i$ by $1$ .

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

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

Factor out $i^{16}$ .

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

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

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

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

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

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

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

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

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

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

One to any power is one.

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

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

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

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

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

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

Factor out $i^{28}$ .

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

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

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

Factor out $i^{2}$ .

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

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

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

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

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

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

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

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

One to any power is one.

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

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

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

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

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

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

$i - 1 - i + n = 0$

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

Subtract $i$ from $i$ .

$0 - 1 + n = 0$

Subtract $1$ from $0$ .

$- 1 + n = 0$

Add $1$ to both sides of the equation.

$n = 1$

Solve for n i^13+i^18+i^31+n=0

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