Basic Math Examples

$\frac{\frac{1}{a + 1} + 1}{\frac{3}{a - 1} + 1}$

Step 1

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

Tap for more steps...

Step 1.1

Multiply $\frac{\frac{1}{a + 1} + 1}{\frac{3}{a - 1} + 1}$ by $\frac{(a + 1) (a - 1)}{(a + 1) (a - 1)}$ .

$\frac{(a + 1) (a - 1)}{(a + 1) (a - 1)} \cdot \frac{\frac{1}{a + 1} + 1}{\frac{3}{a - 1} + 1}$

Step 1.2

Combine.

$\frac{(a + 1) (a - 1) (\frac{1}{a + 1} + 1)}{(a + 1) (a - 1) (\frac{3}{a - 1} + 1)}$

Step 2

Apply the distributive property.

$\frac{(a + 1) (a - 1) \frac{1}{a + 1} + (a + 1) (a - 1) \cdot 1}{(a + 1) (a - 1) \frac{3}{a - 1} + (a + 1) (a - 1) \cdot 1}$

Step 3

Simplify by cancelling.

Tap for more steps...

Step 3.1

Cancel the common factor of $a + 1$ .

Tap for more steps...

Step 3.1.1

Cancel the common factor.

$\frac{(a + 1) (a - 1) \frac{1}{a + 1} + (a + 1) (a - 1) \cdot 1}{(a + 1) (a - 1) \frac{3}{a - 1} + (a + 1) (a - 1) \cdot 1}$

Step 3.1.2

Rewrite the expression.

$\frac{a - 1 + (a + 1) (a - 1) \cdot 1}{(a + 1) (a - 1) \frac{3}{a - 1} + (a + 1) (a - 1) \cdot 1}$

Step 3.2

Cancel the common factor of $a - 1$ .

Tap for more steps...

Step 3.2.1

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

$\frac{a - 1 + (a + 1) (a - 1) \cdot 1}{(a - 1) (a + 1) \frac{3}{a - 1} + (a + 1) (a - 1) \cdot 1}$

Step 3.2.2

Cancel the common factor.

$\frac{a - 1 + (a + 1) (a - 1) \cdot 1}{(a - 1) (a + 1) \frac{3}{a - 1} + (a + 1) (a - 1) \cdot 1}$

Step 3.2.3

Rewrite the expression.

$\frac{a - 1 + (a + 1) (a - 1) \cdot 1}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4

Simplify the numerator.

Tap for more steps...

Step 4.1

Expand $(a + 1) (a - 1)$ using the FOIL Method.

Tap for more steps...

Step 4.1.1

Apply the distributive property.

$\frac{a - 1 + (a (a - 1) + 1 (a - 1)) \cdot 1}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4.1.2

Apply the distributive property.

$\frac{a - 1 + (a \cdot a + a \cdot - 1 + 1 (a - 1)) \cdot 1}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4.1.3

Apply the distributive property.

$\frac{a - 1 + (a \cdot a + a \cdot - 1 + 1 a + 1 \cdot - 1) \cdot 1}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4.2

Simplify and combine like terms.

Tap for more steps...

Step 4.2.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1

Multiply $a$ by $a$ .

$\frac{a - 1 + (a^{2} + a \cdot - 1 + 1 a + 1 \cdot - 1) \cdot 1}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4.2.1.2

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

$\frac{a - 1 + (a^{2} - 1 \cdot a + 1 a + 1 \cdot - 1) \cdot 1}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4.2.1.3

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

$\frac{a - 1 + (a^{2} - a + 1 a + 1 \cdot - 1) \cdot 1}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4.2.1.4

Multiply $a$ by $1$ .

$\frac{a - 1 + (a^{2} - a + a + 1 \cdot - 1) \cdot 1}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4.2.1.5

Multiply $- 1$ by $1$ .

$\frac{a - 1 + (a^{2} - a + a - 1) \cdot 1}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4.2.2

Add $- a$ and $a$ .

$\frac{a - 1 + (a^{2} + 0 - 1) \cdot 1}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4.2.3

Add $a^{2}$ and $0$ .

$\frac{a - 1 + (a^{2} - 1) \cdot 1}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4.3

Multiply $a^{2} - 1$ by $1$ .

$\frac{a - 1 + a^{2} - 1}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4.4

Subtract $1$ from $- 1$ .

$\frac{a + a^{2} - 2}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4.5

Reorder terms.

$\frac{a^{2} + a - 2}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 4.6

Factor $a^{2} + a - 2$ using the AC method.

Tap for more steps...

Step 4.6.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 4.6.2

Write the factored form using these integers.

$\frac{(a - 1) (a + 2)}{(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1}$

Step 5

Simplify the denominator.

Tap for more steps...

Step 5.1

Factor $a + 1$ out of $(a + 1) \cdot 3 + (a + 1) (a - 1) \cdot 1$ .

Tap for more steps...

Step 5.1.1

Factor $a + 1$ out of $(a + 1) \cdot 3$ .

$\frac{(a - 1) (a + 2)}{(a + 1) (3) + (a + 1) (a - 1) \cdot 1}$

Step 5.1.2

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

$\frac{(a - 1) (a + 2)}{(a + 1) (3) + (a + 1) ((a - 1) \cdot 1)}$

Step 5.1.3

Factor $a + 1$ out of $(a + 1) (3) + (a + 1) ((a - 1) \cdot 1)$ .

$\frac{(a - 1) (a + 2)}{(a + 1) (3 + (a - 1) \cdot 1)}$

Step 5.2

Multiply $a - 1$ by $1$ .

$\frac{(a - 1) (a + 2)}{(a + 1) (3 + a - 1)}$

Step 5.3

Subtract $1$ from $3$ .

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

Step 6

Cancel the common factor of $a + 2$ .

Tap for more steps...

Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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