Simplify i^-19

$i^{- 19}$

Step 1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{i^{19}}$

Step 2

Simplify the denominator.

Rewrite $i^{19}$ as ${(i^{4})}^{4} (i^{2} \cdot i)$ .

Factor out $i^{16}$ .

$\frac{1}{i^{16} i^{3}}$

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

$\frac{1}{{(i^{4})}^{4} i^{3}}$

Factor out $i^{2}$ .

$\frac{1}{{(i^{4})}^{4} (i^{2} \cdot i)}$

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

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

$\frac{1}{{({(i^{2})}^{2})}^{4} (i^{2} \cdot i)}$

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

$\frac{1}{{({(- 1)}^{2})}^{4} (i^{2} \cdot i)}$

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

$\frac{1}{1^{4} (i^{2} \cdot i)}$

One to any power is one.

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

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

$\frac{1}{i^{2} \cdot i}$

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

$\frac{1}{- 1 \cdot i}$

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

$\frac{1}{- i}$

Step 3

Cancel the common factor of $1$ and $- 1$ .

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- i}$

Move the negative in front of the fraction.

$- \frac{1}{i}$