Simplify 2i^68+i^71+3i^40+6i^9

$2 i^{68} + i^{71} + 3 i^{40} + 6 i^{9}$

Simplify each term.

Rewrite $i^{68}$ as ${(i^{4})}^{17}$ .

$2 {(i^{4})}^{17} + i^{71} + 3 i^{40} + 6 i^{9}$

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

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

$2 {({(i^{2})}^{2})}^{17} + i^{71} + 3 i^{40} + 6 i^{9}$

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

$2 {({(- 1)}^{2})}^{17} + i^{71} + 3 i^{40} + 6 i^{9}$

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

$2 \cdot 1^{17} + i^{71} + 3 i^{40} + 6 i^{9}$

One to any power is one.

$2 \cdot 1 + i^{71} + 3 i^{40} + 6 i^{9}$

Multiply $2$ by $1$ .

$2 + i^{71} + 3 i^{40} + 6 i^{9}$

Rewrite $i^{71}$ as ${(i^{4})}^{17} (i^{2} \cdot i)$ .

Factor out $i^{68}$ .

$2 + i^{68} i^{3} + 3 i^{40} + 6 i^{9}$

Rewrite $i^{68}$ as ${(i^{4})}^{17}$ .

$2 + {(i^{4})}^{17} i^{3} + 3 i^{40} + 6 i^{9}$

Factor out $i^{2}$ .

$2 + {(i^{4})}^{17} (i^{2} \cdot i) + 3 i^{40} + 6 i^{9}$

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

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

$2 + {({(i^{2})}^{2})}^{17} (i^{2} \cdot i) + 3 i^{40} + 6 i^{9}$

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

$2 + {({(- 1)}^{2})}^{17} (i^{2} \cdot i) + 3 i^{40} + 6 i^{9}$

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

$2 + 1^{17} (i^{2} \cdot i) + 3 i^{40} + 6 i^{9}$

One to any power is one.

$2 + 1 (i^{2} \cdot i) + 3 i^{40} + 6 i^{9}$

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

$2 + i^{2} \cdot i + 3 i^{40} + 6 i^{9}$

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

$2 - 1 \cdot i + 3 i^{40} + 6 i^{9}$

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

$2 - i + 3 i^{40} + 6 i^{9}$

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

$2 - i + 3 {(i^{4})}^{10} + 6 i^{9}$

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

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

$2 - i + 3 {({(i^{2})}^{2})}^{10} + 6 i^{9}$

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

$2 - i + 3 {({(- 1)}^{2})}^{10} + 6 i^{9}$

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

$2 - i + 3 \cdot 1^{10} + 6 i^{9}$

One to any power is one.

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

Multiply $3$ by $1$ .

$2 - i + 3 + 6 i^{9}$

Rewrite $i^{9}$ as ${(i^{4})}^{2} i$ .

Factor out $i^{8}$ .

$2 - i + 3 + 6 (i^{8} i)$

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

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

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

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

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

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

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

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

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

One to any power is one.

$2 - i + 3 + 6 (1 i)$

Multiply $i$ by $1$ .

$2 - i + 3 + 6 i$

Simplify by adding terms.

Add $2$ and $3$ .

$5 - i + 6 i$

Add $- i$ and $6 i$ .

$5 + 5 i$

Simplify 2i^68+i^71+3i^40+6i^9

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