Solve (a+1/a*1/a^2-1+1/a^2+3a*a+3/a+1)*frac{a+1}{2}

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

Evaluate

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

Short Solution Steps

Multiply \frac{a+1}{a} times \frac{1}{a^{2}-1} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{a+1}{a\left(a^{2}-1\right)}+\frac{1}{a^{2}+3a}\times \frac{a+3}{a+1}\right)\times \frac{a+1}{2}$$

Multiply \frac{1}{a^{2}+3a} times \frac{a+3}{a+1} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{a+1}{a\left(a^{2}-1\right)}+\frac{a+3}{\left(a^{2}+3a\right)\left(a+1\right)}\right)\times \frac{a+1}{2}$$

Factor a\left(a^{2}-1\right). Factor \left(a^{2}+3a\right)\left(a+1\right).

$$\left(\frac{a+1}{a\left(a-1\right)\left(a+1\right)}+\frac{a+3}{a\left(a+1\right)\left(a+3\right)}\right)\times \frac{a+1}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-1\right)\left(a+1\right) and a\left(a+1\right)\left(a+3\right) is a\left(a-1\right)\left(a+1\right)\left(a+3\right). Multiply \frac{a+1}{a\left(a-1\right)\left(a+1\right)} times \frac{a+3}{a+3}. Multiply \frac{a+3}{a\left(a+1\right)\left(a+3\right)} times \frac{a-1}{a-1}.

$$\left(\frac{\left(a+1\right)\left(a+3\right)}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}+\frac{\left(a+3\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}\right)\times \frac{a+1}{2}$$

Since \frac{\left(a+1\right)\left(a+3\right)}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)} and \frac{\left(a+3\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(a+1\right)\left(a+3\right)+\left(a+3\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}\times \frac{a+1}{2}$$

Do the multiplications in \left(a+1\right)\left(a+3\right)+\left(a+3\right)\left(a-1\right).

$$\frac{a^{2}+3a+a+3+a^{2}-a+3a-3}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}\times \frac{a+1}{2}$$

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

$$\frac{2a^{2}+6a}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}\times \frac{a+1}{2}$$

Factor the expressions that are not already factored in \frac{2a^{2}+6a}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}.

$$\frac{2a\left(a+3\right)}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}\times \frac{a+1}{2}$$

Cancel out a\left(a+3\right) in both numerator and denominator.

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

Multiply \frac{2}{\left(a-1\right)\left(a+1\right)} times \frac{a+1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2\left(a+1\right) in both numerator and denominator.

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

Expand

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

Short Solution Steps

Multiply \frac{a+1}{a} times \frac{1}{a^{2}-1} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{a+1}{a\left(a^{2}-1\right)}+\frac{1}{a^{2}+3a}\times \frac{a+3}{a+1}\right)\times \frac{a+1}{2}$$

Multiply \frac{1}{a^{2}+3a} times \frac{a+3}{a+1} by multiplying numerator times numerator and denominator times denominator.

$$\left(\frac{a+1}{a\left(a^{2}-1\right)}+\frac{a+3}{\left(a^{2}+3a\right)\left(a+1\right)}\right)\times \frac{a+1}{2}$$

Factor a\left(a^{2}-1\right). Factor \left(a^{2}+3a\right)\left(a+1\right).

$$\left(\frac{a+1}{a\left(a-1\right)\left(a+1\right)}+\frac{a+3}{a\left(a+1\right)\left(a+3\right)}\right)\times \frac{a+1}{2}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of a\left(a-1\right)\left(a+1\right) and a\left(a+1\right)\left(a+3\right) is a\left(a-1\right)\left(a+1\right)\left(a+3\right). Multiply \frac{a+1}{a\left(a-1\right)\left(a+1\right)} times \frac{a+3}{a+3}. Multiply \frac{a+3}{a\left(a+1\right)\left(a+3\right)} times \frac{a-1}{a-1}.

$$\left(\frac{\left(a+1\right)\left(a+3\right)}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}+\frac{\left(a+3\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}\right)\times \frac{a+1}{2}$$

Since \frac{\left(a+1\right)\left(a+3\right)}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)} and \frac{\left(a+3\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(a+1\right)\left(a+3\right)+\left(a+3\right)\left(a-1\right)}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}\times \frac{a+1}{2}$$

Do the multiplications in \left(a+1\right)\left(a+3\right)+\left(a+3\right)\left(a-1\right).

$$\frac{a^{2}+3a+a+3+a^{2}-a+3a-3}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}\times \frac{a+1}{2}$$

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

$$\frac{2a^{2}+6a}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}\times \frac{a+1}{2}$$

Factor the expressions that are not already factored in \frac{2a^{2}+6a}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}.

$$\frac{2a\left(a+3\right)}{a\left(a-1\right)\left(a+1\right)\left(a+3\right)}\times \frac{a+1}{2}$$

Cancel out a\left(a+3\right) in both numerator and denominator.

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

Multiply \frac{2}{\left(a-1\right)\left(a+1\right)} times \frac{a+1}{2} by multiplying numerator times numerator and denominator times denominator.

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

Cancel out 2\left(a+1\right) in both numerator and denominator.

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