Simplify (a/b-b/a)(a-(a^2)/(a+b))

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

Step 1

To write $a$ as a fraction with a common denominator, multiply by $\frac{a + b}{a + b}$ .

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

Step 2

Combine the numerators over the common denominator.

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

Step 3

Simplify the numerator.

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

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

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

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

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

Subtract $a$ from $a$ .

$(\frac{a}{b} - \frac{b}{a}) \frac{a (b + 0)}{a + b}$

Add $b$ and $0$ .

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

Step 4

Multiply $\frac{a}{b} - \frac{b}{a}$ by $\frac{a b}{a + b}$ .

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

Step 5

Simplify the numerator.

To write $\frac{a}{b}$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

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

To write $- \frac{b}{a}$ as a fraction with a common denominator, multiply by $\frac{b}{b}$ .

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

Write each expression with a common denominator of $b a$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{a}{b}$ by $\frac{a}{a}$ .

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

Multiply $\frac{b}{a}$ by $\frac{b}{b}$ .

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

Reorder the factors of $b a$ .

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

Combine the numerators over the common denominator.

$\frac{\frac{a \cdot a - b \cdot b}{a b} a b}{a + b}$

Simplify the numerator.

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

$\frac{\frac{a^{1} a - b \cdot b}{a b} a b}{a + b}$

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

$\frac{\frac{a^{1} a^{1} - b \cdot b}{a b} a b}{a + b}$

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

$\frac{\frac{a^{1 + 1} - b \cdot b}{a b} a b}{a + b}$

Add $1$ and $1$ .

$\frac{\frac{a^{2} - b \cdot b}{a b} a b}{a + b}$

Rewrite $b \cdot b$ as $b^{2}$ .

$\frac{\frac{a^{2} - b^{2}}{a b} a b}{a + b}$

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

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

Combine exponents.

Combine $\frac{(a + b) (a - b)}{a b}$ and $a$ .

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

Combine $\frac{(a + b) (a - b) a}{a b}$ and $b$ .

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

Reduce the expression $\frac{(a + b) (a - b) a b}{a b}$ by cancelling the common factors.

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $b$ .

Cancel the common factor.

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

Divide $(a + b) (a - b)$ by $1$ .

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

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Combine the opposite terms in $a \cdot a + a (- b) + b a + b (- b)$ .

Reorder the factors in the terms $a (- b)$ and $b a$ .

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

Add $- a b$ and $a b$ .

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

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

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

Simplify each term.

Multiply $a$ by $a$ .

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

Rewrite using the commutative property of multiplication.

$\frac{a^{2} - b \cdot b}{a + b}$

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

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

Multiply $b$ by $b$ .

$\frac{a^{2} - b^{2}}{a + b}$

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

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

Step 6

Cancel the common factor of $a + b$ .

Cancel the common factor.

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

Divide $a - b$ by $1$ .

$a - b$

Simplify (a/b-b/a)(a-(a^2)/(a+b))

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