Algebra Examples

$\frac{i^{4} - i^{2}}{i^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

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

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

$\frac{1 - i^{2}}{i^{3}}$

Step 1.2

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

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

Step 1.3

Multiply $- 1$ by $- 1$ .

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

Step 1.4

Add $1$ and $1$ .

$\frac{2}{i^{3}}$

Step 2

Simplify the denominator.

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Step 2.1

Factor out $i^{2}$ .

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

Step 2.2

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

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

Step 2.3

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

$\frac{2}{- i}$

Step 3

Multiply the numerator and denominator of $\frac{2}{- i}$ by the conjugate of $- i$ to make the denominator real.

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

Step 4

Multiply.

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Step 4.1

Combine.

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

Step 4.2

Simplify the denominator.

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Step 4.2.1

Add parentheses.

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

Step 4.2.2

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

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

Step 4.2.3

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

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

Step 4.2.4

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

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

Step 4.2.5

Add $1$ and $1$ .

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

Step 4.2.6

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

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

Step 5

Multiply $- 1$ by $- 1$ .

$\frac{2 i}{1}$

Step 6

Divide $2 i$ by $1$ .

$2 i$