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

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

Step 1

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

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

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

Combine.

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

Step 2

Apply the distributive property.

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

Step 3

Simplify by cancelling.

Cancel the common factor of $1 - a$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $a$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $a$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

Factor $- 1$ out of $- a$ .

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

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

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

Reorder terms.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Cancel the common factor of $1 + a$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Step 4

Simplify the numerator.

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

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

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

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

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

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

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $1$ by $1$ .

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

Multiply $a$ by $1$ .

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

Multiply $- 1$ by $1$ .

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

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

Move $a$ .

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

Multiply $a$ by $a$ .

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

Subtract $a$ from $a$ .

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

Add $1$ and $0$ .

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

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

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

Add $0$ and $1$ .

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

Step 5

Simplify the denominator.

Rewrite ${(a - 1)}^{2}$ as $(a - 1) (a - 1)$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $a$ by $a$ .

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

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

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

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

Subtract $a$ from $- a$ .

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

Apply the distributive property.

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

Simplify.

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

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

Multiply $- 2$ by $- 1$ .

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

Multiply $- 1$ by $1$ .

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

Expand $(- a^{2} + 2 a - 1) (1 + a)$ by multiplying each term in the first expression by each term in the second expression.

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

Simplify each term.

Multiply $- 1$ by $1$ .

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

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

Move $a$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $2$ by $1$ .

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

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

Move $a$ .

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

Multiply $a$ by $a$ .

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

Multiply $- 1$ by $1$ .

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

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

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

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

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

Subtract $a$ from $2 a$ .

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

Apply the distributive property.

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

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

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

Rewrite using the commutative property of multiplication.

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

Simplify each term.

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

Move $a^{2}$ .

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

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

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

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

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

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

Add $2$ and $1$ .

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

Multiply $- 1$ by $- 1$ .

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

Multiply $a^{3}$ by $1$ .

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

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

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

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

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

Add $- a^{3}$ and $a^{3}$ .

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

Add $a - 1$ and $0$ .

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

Step 6

Multiply $1 + a$ by $1$ .

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

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

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