Solve (a/3a-3-3/a^2+3a+a^2+9/27-3a^2)

$$( \frac { a } { 3 a - 3 } - \frac { 3 } { a ^ { 2 } + 3 a } + \frac { a ^ { 2 } + 9 } { 27 - 3 a ^ { 2 } } )$$

Evaluate

$$\frac{27-45a+27a^{2}-a^{3}}{3a\left(a-1\right)\left(9-a^{2}\right)}$$

Short Solution Steps

Factor 3a-3. Factor a^{2}+3a.

$$\frac{a}{3\left(a-1\right)}-\frac{3}{a\left(a+3\right)}+\frac{a^{2}+9}{27-3a^{2}}$$

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

$$\frac{aa\left(a+3\right)}{3a\left(a-1\right)\left(a+3\right)}-\frac{3\times 3\left(a-1\right)}{3a\left(a-1\right)\left(a+3\right)}+\frac{a^{2}+9}{27-3a^{2}}$$

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

$$\frac{aa\left(a+3\right)-3\times 3\left(a-1\right)}{3a\left(a-1\right)\left(a+3\right)}+\frac{a^{2}+9}{27-3a^{2}}$$

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

$$\frac{a^{3}+3a^{2}-9a+9}{3a\left(a-1\right)\left(a+3\right)}+\frac{a^{2}+9}{27-3a^{2}}$$

Factor 27-3a^{2}.

$$\frac{a^{3}+3a^{2}-9a+9}{3a\left(a-1\right)\left(a+3\right)}+\frac{a^{2}+9}{3\left(a-3\right)\left(-a-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3a\left(a-1\right)\left(a+3\right) and 3\left(a-3\right)\left(-a-3\right) is 3a\left(a-3\right)\left(a-1\right)\left(a+3\right). Multiply \frac{a^{3}+3a^{2}-9a+9}{3a\left(a-1\right)\left(a+3\right)} times \frac{a-3}{a-3}. Multiply \frac{a^{2}+9}{3\left(a-3\right)\left(-a-3\right)} times \frac{-a\left(a-1\right)}{-a\left(a-1\right)}.

$$\frac{\left(a^{3}+3a^{2}-9a+9\right)\left(a-3\right)}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)}+\frac{\left(a^{2}+9\right)\left(-1\right)a\left(a-1\right)}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)}$$

Since \frac{\left(a^{3}+3a^{2}-9a+9\right)\left(a-3\right)}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)} and \frac{\left(a^{2}+9\right)\left(-1\right)a\left(a-1\right)}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(a^{3}+3a^{2}-9a+9\right)\left(a-3\right)+\left(a^{2}+9\right)\left(-1\right)a\left(a-1\right)}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)}$$

Do the multiplications in \left(a^{3}+3a^{2}-9a+9\right)\left(a-3\right)+\left(a^{2}+9\right)\left(-1\right)a\left(a-1\right).

$$\frac{a^{4}-3a^{3}+3a^{3}-9a^{2}-9a^{2}+27a+9a-27-a^{4}+a^{3}-9a^{2}+9a}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)}$$

Combine like terms in a^{4}-3a^{3}+3a^{3}-9a^{2}-9a^{2}+27a+9a-27-a^{4}+a^{3}-9a^{2}+9a.

$$\frac{a^{3}-27a^{2}+45a-27}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)}$$

Expand 3a\left(a-3\right)\left(a-1\right)\left(a+3\right).

$$\frac{a^{3}-27a^{2}+45a-27}{3a^{4}-3a^{3}-27a^{2}+27a}$$

Expand

$$\frac{a^{3}-27a^{2}+45a-27}{3a\left(a-1\right)\left(a^{2}-9\right)}$$

Short Solution Steps

Factor 3a-3. Factor a^{2}+3a.

$$\frac{a}{3\left(a-1\right)}-\frac{3}{a\left(a+3\right)}+\frac{a^{2}+9}{27-3a^{2}}$$

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

$$\frac{aa\left(a+3\right)}{3a\left(a-1\right)\left(a+3\right)}-\frac{3\times 3\left(a-1\right)}{3a\left(a-1\right)\left(a+3\right)}+\frac{a^{2}+9}{27-3a^{2}}$$

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

$$\frac{aa\left(a+3\right)-3\times 3\left(a-1\right)}{3a\left(a-1\right)\left(a+3\right)}+\frac{a^{2}+9}{27-3a^{2}}$$

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

$$\frac{a^{3}+3a^{2}-9a+9}{3a\left(a-1\right)\left(a+3\right)}+\frac{a^{2}+9}{27-3a^{2}}$$

Factor 27-3a^{2}.

$$\frac{a^{3}+3a^{2}-9a+9}{3a\left(a-1\right)\left(a+3\right)}+\frac{a^{2}+9}{3\left(a-3\right)\left(-a-3\right)}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3a\left(a-1\right)\left(a+3\right) and 3\left(a-3\right)\left(-a-3\right) is 3a\left(a-3\right)\left(a-1\right)\left(a+3\right). Multiply \frac{a^{3}+3a^{2}-9a+9}{3a\left(a-1\right)\left(a+3\right)} times \frac{a-3}{a-3}. Multiply \frac{a^{2}+9}{3\left(a-3\right)\left(-a-3\right)} times \frac{-a\left(a-1\right)}{-a\left(a-1\right)}.

$$\frac{\left(a^{3}+3a^{2}-9a+9\right)\left(a-3\right)}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)}+\frac{\left(a^{2}+9\right)\left(-1\right)a\left(a-1\right)}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)}$$

Since \frac{\left(a^{3}+3a^{2}-9a+9\right)\left(a-3\right)}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)} and \frac{\left(a^{2}+9\right)\left(-1\right)a\left(a-1\right)}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)} have the same denominator, add them by adding their numerators.

$$\frac{\left(a^{3}+3a^{2}-9a+9\right)\left(a-3\right)+\left(a^{2}+9\right)\left(-1\right)a\left(a-1\right)}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)}$$

Do the multiplications in \left(a^{3}+3a^{2}-9a+9\right)\left(a-3\right)+\left(a^{2}+9\right)\left(-1\right)a\left(a-1\right).

$$\frac{a^{4}-3a^{3}+3a^{3}-9a^{2}-9a^{2}+27a+9a-27-a^{4}+a^{3}-9a^{2}+9a}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)}$$

Combine like terms in a^{4}-3a^{3}+3a^{3}-9a^{2}-9a^{2}+27a+9a-27-a^{4}+a^{3}-9a^{2}+9a.

$$\frac{a^{3}-27a^{2}+45a-27}{3a\left(a-3\right)\left(a-1\right)\left(a+3\right)}$$

Expand 3a\left(a-3\right)\left(a-1\right)\left(a+3\right).

$$\frac{a^{3}-27a^{2}+45a-27}{3a^{4}-3a^{3}-27a^{2}+27a}$$