Write in Standard Form i^-15

$i^{- 15}$

Step 1

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

$\frac{1}{i^{15}}$

Step 2

Simplify the denominator.

Rewrite $i^{15}$ as ${(i^{4})}^{3} (i^{2} \cdot i)$ .

Factor out $i^{12}$ .

$\frac{1}{i^{12} i^{3}}$

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

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

Factor out $i^{2}$ .

$\frac{1}{{(i^{4})}^{3} (i^{2} \cdot i)}$

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

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

$\frac{1}{{({(i^{2})}^{2})}^{3} (i^{2} \cdot i)}$

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

$\frac{1}{{({(- 1)}^{2})}^{3} (i^{2} \cdot i)}$

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

$\frac{1}{1^{3} (i^{2} \cdot i)}$

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.

$- \frac{1}{i}$

Step 4

Multiply the numerator and denominator of $\frac{1}{i}$ by the conjugate of $i$ to make the denominator real.

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

Step 5

Multiply.

Combine.

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

Multiply $i$ by $1$ .

$- \frac{i}{i i}$

Simplify the denominator.

Raise $i$ to the power of $1$ .

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

Raise $i$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{i}{i^{1 + 1}}$

Add $1$ and $1$ .

$- \frac{i}{i^{2}}$

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

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

Step 6

Move the negative one from the denominator of $\frac{i}{- 1}$ .

$- (- 1 \cdot i)$

Step 7

Rewrite $- 1 \cdot i$ as $- i$ .

$- - i$

Step 8

Multiply $- 1$ by $- 1$ .

$1 i$

Step 9

Multiply $i$ by $1$ .

$i$

Write in Standard Form i^-15

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