Precalculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Cancel the common factor of $a$ .

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Step 3.1

Cancel the common factor.

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

Step 3.2

Rewrite the expression.

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

Step 4

Simplify the numerator.

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Step 4.1

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

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

Step 4.2

Rewrite $- 1 a$ as $- a$ .

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

Step 5

Simplify the denominator.

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Step 5.1

Multiply $a$ by $a$ .

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

Step 5.2

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

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

Step 5.3

Rewrite $- 1 a$ as $- a$ .

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

Step 5.4

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

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Step 5.4.1

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

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

Step 5.4.2

Factor $a$ out of $- a$ .

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

Step 5.4.3

Factor $a$ out of $a \cdot a + a \cdot - 1$ .

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

Step 6

Cancel the common factor of $1 - a$ and $a - 1$ .

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Step 6.1

Rewrite $1$ as $- 1 (- 1)$ .

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

Step 6.2

Factor $- 1$ out of $- a$ .

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

Step 6.3

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

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

Step 6.4

Reorder terms.

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

Step 6.5

Cancel the common factor.

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

Step 6.6

Rewrite the expression.

$\frac{- 1}{a}$

Step 7

Move the negative in front of the fraction.

$- \frac{1}{a}$