Algebra Examples

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

Step 1

Move $x^{- 1}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{(x - x^{- 1}) x}$

Step 2

Simplify the denominator.

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Step 2.1

Factor $x^{- 1}$ out of $x - x^{- 1}$ .

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Step 2.1.1

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

$\frac{1}{(x^{1} - x^{- 1}) x}$

Step 2.1.2

Factor $x^{- 1}$ out of $x^{1}$ .

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

Step 2.1.3

Factor $x^{- 1}$ out of $- x^{- 1}$ .

$\frac{1}{(x^{- 1} x^{2} + x^{- 1} \cdot - 1) x}$

Step 2.1.4

Factor $x^{- 1}$ out of $x^{- 1} x^{2} + x^{- 1} \cdot - 1$ .

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

Step 2.2

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

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

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$\frac{1}{x^{- 1} (x + 1) (x - 1) x}$

Step 2.4

Combine exponents.

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Step 2.4.1

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

$\frac{1}{x^{1} x^{- 1} (x + 1) (x - 1)}$

Step 2.4.2

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

$\frac{1}{x^{1 - 1} (x + 1) (x - 1)}$

Step 2.4.3

Subtract $1$ from $1$ .

$\frac{1}{x^{0} (x + 1) (x - 1)}$

Step 2.5

Anything raised to $0$ is $1$ .

$\frac{1}{1 (x + 1) (x - 1)}$

Step 2.6

Multiply $x + 1$ by $1$ .

$\frac{1}{(x + 1) (x - 1)}$