Simplify (a/(a-1)-(1+a)/a)/((a-1)/a-a/(1+a))

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

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

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

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

Combine.

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

Apply the distributive property.

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

Simplify by cancelling.

Cancel the common factor of $a - 1$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $a$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $a$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $1 + a$ .

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

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

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Simplify the numerator.

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

Factor $1 + a$ out of $a^{2} (1 + a)$ .

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

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

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

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

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

Apply the distributive property.

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

Multiply $- 1$ by $1$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

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

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

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

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

Rewrite using the commutative property of multiplication.

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

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

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

Multiply $a$ by $a$ .

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

Multiply $- 1$ by $- 1$ .

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

Multiply $- 1 (- a)$ .

Multiply $- 1$ by $- 1$ .

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

Multiply $a$ by $1$ .

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

Add $- a$ and $a$ .

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

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

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

Subtract $a^{2}$ from $a^{2}$ .

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

Add $0$ and $1$ .

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

Simplify the denominator.

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

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

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Combine the opposite terms in $a \cdot 1 + a \cdot a - 1 \cdot 1 - 1 a$ .

Reorder the factors in the terms $a \cdot 1$ and $- 1 a$ .

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

Subtract $1 a$ from $1 a$ .

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

Add $0$ and $a \cdot a$ .

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

Simplify each term.

Multiply $a$ by $a$ .

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

Multiply $- 1$ by $1$ .

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

Subtract $a^{2}$ from $a^{2}$ .

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

Subtract $1$ from $0$ .

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

Simplify the expression.

Multiply $1 + a$ by $1$ .

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

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

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

Move the negative in front of the fraction.

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

Simplify (a/(a-1)-(1+a)/a)/((a-1)/a-a/(1+a))

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