Simplify i^51

$i^{51}$

Step 1

Rewrite $i^{51}$ as ${(i^{4})}^{12} (i^{2} \cdot i)$ .

Factor out $i^{48}$ .

$i^{48} i^{3}$

Rewrite $i^{48}$ as ${(i^{4})}^{12}$ .

${(i^{4})}^{12} i^{3}$

Factor out $i^{2}$ .

${(i^{4})}^{12} (i^{2} \cdot i)$

Step 2

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

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

${({(i^{2})}^{2})}^{12} (i^{2} \cdot i)$

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

${({(- 1)}^{2})}^{12} (i^{2} \cdot i)$

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

$1^{12} (i^{2} \cdot i)$

Step 3

One to any power is one.

$1 (i^{2} \cdot i)$

Step 4

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

$i^{2} \cdot i$

Step 5

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

$- 1 \cdot i$

Step 6

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

$- i$