Algebra Examples

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

Step 1

Multiply the numerator and denominator of the fraction by $(1 - a) (1 + a)$ .

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Step 1.1

Multiply $\frac{\frac{2}{1 - a} + \frac{2}{1 + a}}{\frac{2}{1 + a} - \frac{2}{1 - a}}$ by $\frac{(1 - a) (1 + a)}{(1 - a) (1 + a)}$ .

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

Step 1.2

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

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Step 3.1

Cancel the common factor of $1 - a$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $1 + a$ .

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Cancel the common factor of $1 + a$ .

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Step 3.3.1

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

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

Step 3.3.2

Cancel the common factor.

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

Step 3.3.3

Rewrite the expression.

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

Step 3.4

Cancel the common factor of $1 - a$ .

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Step 3.4.1

Move the leading negative in $- \frac{2}{1 - a}$ into the numerator.

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

Step 3.4.2

Cancel the common factor.

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

Step 3.4.3

Rewrite the expression.

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

Step 4

Cancel the common factor of $(1 + a) \cdot 2 + (1 - a) \cdot 2$ and $(1 - a) \cdot 2 + (1 + a) \cdot - 2$ .

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Step 4.1

Factor $2$ out of $(1 + a) \cdot 2$ .

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

Step 4.2

Factor $2$ out of $(1 - a) \cdot 2$ .

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

Step 4.3

Factor $2$ out of $2 \cdot (1 + a) + 2 \cdot (1 - a)$ .

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

Step 4.4

Cancel the common factors.

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Step 4.4.1

Factor $2$ out of $(1 - a) \cdot 2$ .

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

Step 4.4.2

Factor $2$ out of $(1 + a) \cdot - 2$ .

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

Step 4.4.3

Factor $2$ out of $2 \cdot (1 - a) + 2 ((1 + a) \cdot - 1)$ .

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

Step 4.4.4

Cancel the common factor.

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

Step 4.4.5

Rewrite the expression.

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

Step 5

Simplify the numerator.

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Step 5.1

Add $1$ and $1$ .

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

Step 5.2

Subtract $a$ from $a$ .

$\frac{0 + 2}{1 - a + (1 + a) \cdot - 1}$

Step 5.3

Add $0$ and $2$ .

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

Step 6

Simplify the denominator.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Multiply $- 1$ by $1$ .

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

Step 6.3

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

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

Step 6.4

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

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

Step 6.5

Subtract $1$ from $1$ .

$\frac{2}{- a + 0 - a}$

Step 6.6

Add $- a$ and $0$ .

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

Step 6.7

Subtract $a$ from $- a$ .

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

Step 7

Reduce the expression by cancelling the common factors.

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Step 7.1

Cancel the common factor of $2$ and $- 2$ .

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Step 7.1.1

Factor $2$ out of $2$ .

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

Step 7.1.2

Cancel the common factors.

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Step 7.1.2.1

Factor $2$ out of $- 2 a$ .

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

Step 7.1.2.2

Cancel the common factor.

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

Step 7.1.2.3

Rewrite the expression.

$\frac{1}{- a}$

Step 7.2

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

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Step 7.2.1

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

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

Step 7.2.2

Move the negative in front of the fraction.

$- \frac{1}{a}$