Precalculus Examples

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

Step 1

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

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Step 1.1

Multiply $\frac{\frac{1}{x} - \frac{1}{a}}{x - a}$ by $\frac{x a}{x a}$ .

$\frac{x a}{x a} \cdot \frac{\frac{1}{x} - \frac{1}{a}}{x - a}$

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

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

Factor $x$ out of $x a$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $a$ .

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Step 3.2.1

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

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

Step 3.2.2

Factor $a$ out of $x a$ .

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

Step 3.2.3

Cancel the common factor.

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

Step 3.2.4

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

Move $- 1$ to the left of $x$ .

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

Step 4.2

Rewrite $- 1 x$ as $- x$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Multiply $x$ by $x$ by adding the exponents.

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Step 5.1.1

Move $x$ .

$\frac{a - x}{x \cdot x a + x a (- a)}$

Step 5.1.2

Multiply $x$ by $x$ .

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

Step 5.2

Rewrite using the commutative property of multiplication.

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

Step 5.3

Multiply $a$ by $a$ by adding the exponents.

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Step 5.3.1

Move $a$ .

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

Step 5.3.2

Multiply $a$ by $a$ .

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

Step 5.4

Factor $x a$ out of $x^{2} a - x a^{2}$ .

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Step 5.4.1

Factor $x a$ out of $x^{2} a$ .

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

Step 5.4.2

Factor $x a$ out of $- x a^{2}$ .

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

Step 5.4.3

Factor $x a$ out of $x a (x) + x a (- a)$ .

$\frac{a - x}{x a (x - a)}$

Step 6

Cancel the common factor of $a - x$ and $x - a$ .

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Step 6.1

Factor $- 1$ out of $a$ .

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

Step 6.2

Factor $- 1$ out of $- x$ .

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

Step 6.3

Factor $- 1$ out of $- 1 (- a) - (x)$ .

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

Step 6.4

Reorder terms.

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

Step 6.5

Cancel the common factor.

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

Step 6.6

Rewrite the expression.

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

Step 7

Move the negative in front of the fraction.

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