Solve 2T/m^2-81+3/2(m+9)

$$\frac { 2 T } { m ^ { 2 } - 81 } + \frac { 3 } { 2 ( m + 9 ) }$$

Evaluate

$$\frac{3m+4T-27}{2\left(m^{2}-81\right)}$$

$$|m|\neq 9$$

Short Solution Steps

Factor m^{2}-81.

$$\frac{2T}{\left(m-9\right)\left(m+9\right)}+\frac{3}{2\left(m+9\right)}$$

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

$$\frac{2\times 2T}{2\left(m-9\right)\left(m+9\right)}+\frac{3\left(m-9\right)}{2\left(m-9\right)\left(m+9\right)}$$

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

$$\frac{2\times 2T+3\left(m-9\right)}{2\left(m-9\right)\left(m+9\right)}$$

Do the multiplications in 2\times 2T+3\left(m-9\right).

$$\frac{4T+3m-27}{2\left(m-9\right)\left(m+9\right)}$$

Expand 2\left(m-9\right)\left(m+9\right).

$$\frac{4T+3m-27}{2m^{2}-162}$$

Expand

$$\frac{3m+4T-27}{2\left(m^{2}-81\right)}$$

$$|m|\neq 9$$

Short Solution Steps

Factor m^{2}-81.

$$\frac{2T}{\left(m-9\right)\left(m+9\right)}+\frac{3}{2\left(m+9\right)}$$

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

$$\frac{2\times 2T}{2\left(m-9\right)\left(m+9\right)}+\frac{3\left(m-9\right)}{2\left(m-9\right)\left(m+9\right)}$$

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

$$\frac{2\times 2T+3\left(m-9\right)}{2\left(m-9\right)\left(m+9\right)}$$

Do the multiplications in 2\times 2T+3\left(m-9\right).

$$\frac{4T+3m-27}{2\left(m-9\right)\left(m+9\right)}$$

Expand 2\left(m-9\right)\left(m+9\right).

$$\frac{4T+3m-27}{2m^{2}-162}$$