Simplify (1+1/(x-1))/(1+1/(x^2-1))

$\frac{1 + \frac{1}{x - 1}}{1 + \frac{1}{x^{2} - 1}}$

Step 1

Multiply the numerator and denominator of the complex fraction by $(x - 1) (x^{2} - 1)$ .

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

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

Combine.

$\frac{(x - 1) (x^{2} - 1) (1 + \frac{1}{x - 1})}{(x - 1) (x^{2} - 1) (1 + \frac{1}{x^{2} - 1})}$

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x^{2} - 1$ .

Factor $x^{2} - 1$ out of $(x - 1) (x^{2} - 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 4

Simplify the numerator.

Expand $(x - 1) (x^{2} - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

Multiply $x$ by $x^{2}$ by adding the exponents.

Multiply $x$ by $x^{2}$ .

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

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

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

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

Add $1$ and $2$ .

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

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

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

Rewrite $- 1 x$ as $- x$ .

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

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x^{3} - x - x^{2} + 1$ by $1$ .

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

Add $- x^{2}$ and $x^{2}$ .

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

Add $x^{3}$ and $0$ .

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

Subtract $1$ from $1$ .

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

Add $x^{3} - x$ and $0$ .

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

Rewrite $x^{3} - x$ in a factored form.

Factor $x$ out of $x^{3} - x$ .

Factor $x$ out of $x^{3}$ .

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

Factor $x$ out of $- x$ .

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

Factor $x$ out of $x \cdot x^{2} + x \cdot - 1$ .

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

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

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

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

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

Step 5

Simplify the denominator.

Expand $(x - 1) (x^{2} - 1)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify each term.

Multiply $x$ by $x^{2}$ by adding the exponents.

Multiply $x$ by $x^{2}$ .

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

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

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

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

Add $1$ and $2$ .

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

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

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

Rewrite $- 1 x$ as $- x$ .

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

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

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

Multiply $- 1$ by $- 1$ .

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

Multiply $x^{3} - x - x^{2} + 1$ by $1$ .

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

Add $- x$ and $x$ .

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

Add $x^{3}$ and $0$ .

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

Subtract $1$ from $1$ .

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

Add $x^{3} - x^{2}$ and $0$ .

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

Factor $x^{2}$ out of $x^{3} - x^{2}$ .

Factor $x^{2}$ out of $x^{3}$ .

$\frac{x (x + 1) (x - 1)}{x^{2} x - x^{2}}$

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

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

Factor $x^{2}$ out of $x^{2} x + x^{2} \cdot - 1$ .

$\frac{x (x + 1) (x - 1)}{x^{2} (x - 1)}$

Step 6

Reduce the expression by cancelling the common factors.

Cancel the common factor of $x$ and $x^{2}$ .

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

$\frac{x ((x + 1) (x - 1))}{x^{2} (x - 1)}$

Cancel the common factors.

Factor $x$ out of $x^{2} (x - 1)$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $x - 1$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{x + 1}{x}$

Simplify (1+1/(x-1))/(1+1/(x^2-1))

Related Questions

Support LCMGCF.com by disabling your adblocker.