Algebra Examples

$i^{- 9}$

Step 1

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

$\frac{1}{i^{9}}$

Step 2

Simplify the denominator.

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Step 2.1

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

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Step 2.1.1

Factor out $i^{8}$ .

$\frac{1}{i^{8} i}$

Step 2.1.2

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

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

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

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

Step 2.3

One to any power is one.

$\frac{1}{1 i}$

Step 2.4

Multiply $i$ by $1$ .

$\frac{1}{i}$

Step 3

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 4

Multiply.

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Step 4.1

Combine.

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

Step 4.2

Multiply $i$ by $1$ .

$\frac{i}{i i}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

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

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

Step 4.3.4

Add $1$ and $1$ .

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

Step 4.3.5

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

$\frac{i}{- 1}$

Step 5

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

$- 1 \cdot i$

Step 6

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

$- i$