Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

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

Factor $b$ out of $b a$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Add $1$ and $1$ .

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

Step 3.6

Cancel the common factor of $a$ .

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Step 3.6.1

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

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

Step 3.6.2

Factor $a$ out of $b a$ .

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

Step 3.6.3

Cancel the common factor.

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

Step 3.6.4

Rewrite the expression.

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Add $1$ and $1$ .

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

Step 3.11

Cancel the common factor of $a$ .

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Step 3.11.1

Factor $a$ out of $b a$ .

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

Step 3.11.2

Cancel the common factor.

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

Step 3.11.3

Rewrite the expression.

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

Step 3.12

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

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

Step 3.13

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

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

Step 3.14

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

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

Step 3.15

Add $1$ and $1$ .

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

Step 4

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

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

Step 5

Simplify the denominator.

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Step 5.1

Factor $b$ out of $b a \cdot 1 + b^{2}$ .

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Step 5.1.1

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

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

Step 5.1.2

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

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

Step 5.1.3

Factor $b$ out of $b (a \cdot 1) + b \cdot b$ .

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

Step 5.2

Multiply $a$ by $1$ .

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

Step 6

Cancel the common factor of $a + b$ .

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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