Solve a+b+1/a+b-(a+1/a)/b

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

Evaluate

$$\frac{a^{2}+ab-1}{a\left(a+b\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a}{a}.

$$\frac{a+b+\frac{1}{a+b}-\left(\frac{aa}{a}+\frac{1}{a}\right)}{b}$$

Since \frac{aa}{a} and \frac{1}{a} have the same denominator, add them by adding their numerators.

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

Do the multiplications in aa+1.

$$\frac{a+b+\frac{1}{a+b}-\frac{a^{2}+1}{a}}{b}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a+b times \frac{a+b}{a+b}.

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

Since \frac{\left(a+b\right)\left(a+b\right)}{a+b} and \frac{1}{a+b} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(a+b\right)\left(a+b\right)+1.

$$\frac{\frac{a^{2}+ab+ab+b^{2}+1}{a+b}-\frac{a^{2}+1}{a}}{b}$$

Combine like terms in a^{2}+ab+ab+b^{2}+1.

$$\frac{\frac{a^{2}+b^{2}+2ab+1}{a+b}-\frac{a^{2}+1}{a}}{b}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+b and a is a\left(a+b\right). Multiply \frac{a^{2}+b^{2}+2ab+1}{a+b} times \frac{a}{a}. Multiply \frac{a^{2}+1}{a} times \frac{a+b}{a+b}.

$$\frac{\frac{\left(a^{2}+b^{2}+2ab+1\right)a}{a\left(a+b\right)}-\frac{\left(a^{2}+1\right)\left(a+b\right)}{a\left(a+b\right)}}{b}$$

Since \frac{\left(a^{2}+b^{2}+2ab+1\right)a}{a\left(a+b\right)} and \frac{\left(a^{2}+1\right)\left(a+b\right)}{a\left(a+b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(a^{2}+b^{2}+2ab+1\right)a-\left(a^{2}+1\right)\left(a+b\right)}{a\left(a+b\right)}}{b}$$

Do the multiplications in \left(a^{2}+b^{2}+2ab+1\right)a-\left(a^{2}+1\right)\left(a+b\right).

$$\frac{\frac{a^{3}+b^{2}a+2a^{2}b+a-a^{3}-a^{2}b-a-b}{a\left(a+b\right)}}{b}$$

Combine like terms in a^{3}+b^{2}a+2a^{2}b+a-a^{3}-a^{2}b-a-b.

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

Express \frac{\frac{-b+a^{2}b+b^{2}a}{a\left(a+b\right)}}{b} as a single fraction.

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

Factor the expressions that are not already factored.

$$\frac{b\left(a^{2}+ab-1\right)}{ab\left(a+b\right)}$$

Cancel out b in both numerator and denominator.

$$\frac{a^{2}+ab-1}{a\left(a+b\right)}$$

Expand the expression.

$$\frac{a^{2}+ab-1}{a^{2}+ab}$$

Expand

$$\frac{a^{2}+ab-1}{a\left(a+b\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{a}{a}.

$$\frac{a+b+\frac{1}{a+b}-\left(\frac{aa}{a}+\frac{1}{a}\right)}{b}$$

Since \frac{aa}{a} and \frac{1}{a} have the same denominator, add them by adding their numerators.

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

Do the multiplications in aa+1.

$$\frac{a+b+\frac{1}{a+b}-\frac{a^{2}+1}{a}}{b}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a+b times \frac{a+b}{a+b}.

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

Since \frac{\left(a+b\right)\left(a+b\right)}{a+b} and \frac{1}{a+b} have the same denominator, add them by adding their numerators.

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

Do the multiplications in \left(a+b\right)\left(a+b\right)+1.

$$\frac{\frac{a^{2}+ab+ab+b^{2}+1}{a+b}-\frac{a^{2}+1}{a}}{b}$$

Combine like terms in a^{2}+ab+ab+b^{2}+1.

$$\frac{\frac{a^{2}+b^{2}+2ab+1}{a+b}-\frac{a^{2}+1}{a}}{b}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a+b and a is a\left(a+b\right). Multiply \frac{a^{2}+b^{2}+2ab+1}{a+b} times \frac{a}{a}. Multiply \frac{a^{2}+1}{a} times \frac{a+b}{a+b}.

$$\frac{\frac{\left(a^{2}+b^{2}+2ab+1\right)a}{a\left(a+b\right)}-\frac{\left(a^{2}+1\right)\left(a+b\right)}{a\left(a+b\right)}}{b}$$

Since \frac{\left(a^{2}+b^{2}+2ab+1\right)a}{a\left(a+b\right)} and \frac{\left(a^{2}+1\right)\left(a+b\right)}{a\left(a+b\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(a^{2}+b^{2}+2ab+1\right)a-\left(a^{2}+1\right)\left(a+b\right)}{a\left(a+b\right)}}{b}$$

Do the multiplications in \left(a^{2}+b^{2}+2ab+1\right)a-\left(a^{2}+1\right)\left(a+b\right).

$$\frac{\frac{a^{3}+b^{2}a+2a^{2}b+a-a^{3}-a^{2}b-a-b}{a\left(a+b\right)}}{b}$$

Combine like terms in a^{3}+b^{2}a+2a^{2}b+a-a^{3}-a^{2}b-a-b.

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

Express \frac{\frac{-b+a^{2}b+b^{2}a}{a\left(a+b\right)}}{b} as a single fraction.

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

Factor the expressions that are not already factored.

$$\frac{b\left(a^{2}+ab-1\right)}{ab\left(a+b\right)}$$

Cancel out b in both numerator and denominator.

$$\frac{a^{2}+ab-1}{a\left(a+b\right)}$$

Expand the expression.

$$\frac{a^{2}+ab-1}{a^{2}+ab}$$