Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $x$ .

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Step 1.1

Multiply $\frac{x^{2} - \frac{1}{x}}{1 - \frac{1}{x}}$ by $\frac{x}{x}$ .

$\frac{x}{x} \cdot \frac{x^{2} - \frac{1}{x}}{1 - \frac{1}{x}}$

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $x$ .

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Step 3.1.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $x$ .

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Step 3.2.1

Move the leading negative in $- \frac{1}{x}$ into the numerator.

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

$\frac{x \cdot x^{2} - 1}{x \cdot 1 - 1}$

Step 4

Simplify the numerator.

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Step 4.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 4.1.1

Multiply $x$ by $x^{2}$ .

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Step 4.1.1.1

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

$\frac{x^{1} x^{2} - 1}{x \cdot 1 - 1}$

Step 4.1.1.2

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

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

Step 4.1.2

Add $1$ and $2$ .

$\frac{x^{3} - 1}{x \cdot 1 - 1}$

Step 4.2

Rewrite $1$ as $1^{3}$ .

$\frac{x^{3} - 1^{3}}{x \cdot 1 - 1}$

Step 4.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = x$ and $b = 1$ .

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

Step 4.4

Simplify.

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Step 4.4.1

Multiply $x$ by $1$ .

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

Step 4.4.2

One to any power is one.

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

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Multiply $x$ by $1$ .

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

Step 5.2

Cancel the common factor of $x - 1$ .

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Step 5.2.1

Cancel the common factor.

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

Step 5.2.2

Divide $x^{2} + x + 1$ by $1$ .

$x^{2} + x + 1$