Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $b$ .

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Step 3.1.1

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

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $b$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 4

Factor $a$ out of $b a - a$ .

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Step 4.1

Factor $a$ out of $b a$ .

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

Step 4.2

Factor $a$ out of $- a$ .

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

Step 4.3

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

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

$\frac{a (b - 1)}{b^{2} - 1}$

Step 5.5

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

$\frac{a (b - 1)}{b^{2} - 1^{2}}$

Step 5.6

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

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

Step 6

Cancel the common factor of $b - 1$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\frac{a}{b + 1}$