Solve (g/g-h-g^2/g^2-h^2)*4g+4h/3g

$$( \frac { g } { g - h } - \frac { g ^ { 2 } } { g ^ { 2 } - h ^ { 2 } } ) \cdot \frac { 4 g + 4 h } { 3 g }$$

Evaluate

$$\frac{4h}{3\left(g-h\right)}$$

Short Solution Steps

Factor g^{2}-h^{2}.

$$\left(\frac{g}{g-h}-\frac{g^{2}}{\left(g+h\right)\left(g-h\right)}\right)\times \frac{4g+4h}{3g}$$

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

$$\left(\frac{g\left(g+h\right)}{\left(g+h\right)\left(g-h\right)}-\frac{g^{2}}{\left(g+h\right)\left(g-h\right)}\right)\times \frac{4g+4h}{3g}$$

Since \frac{g\left(g+h\right)}{\left(g+h\right)\left(g-h\right)} and \frac{g^{2}}{\left(g+h\right)\left(g-h\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{g\left(g+h\right)-g^{2}}{\left(g+h\right)\left(g-h\right)}\times \frac{4g+4h}{3g}$$

Do the multiplications in g\left(g+h\right)-g^{2}.

$$\frac{g^{2}+gh-g^{2}}{\left(g+h\right)\left(g-h\right)}\times \frac{4g+4h}{3g}$$

Combine like terms in g^{2}+gh-g^{2}.

$$\frac{gh}{\left(g+h\right)\left(g-h\right)}\times \frac{4g+4h}{3g}$$

Multiply \frac{gh}{\left(g+h\right)\left(g-h\right)} times \frac{4g+4h}{3g} by multiplying numerator times numerator and denominator times denominator.

$$\frac{gh\left(4g+4h\right)}{\left(g+h\right)\left(g-h\right)\times 3g}$$

Cancel out g in both numerator and denominator.

$$\frac{h\left(4g+4h\right)}{3\left(g+h\right)\left(g-h\right)}$$

Factor the expressions that are not already factored.

$$\frac{4h\left(g+h\right)}{3\left(g+h\right)\left(g-h\right)}$$

Cancel out g+h in both numerator and denominator.

$$\frac{4h}{3\left(g-h\right)}$$

Expand the expression.

$$\frac{4h}{3g-3h}$$

Expand

$$-\frac{4h}{3\left(h-g\right)}$$

Short Solution Steps

Factor g^{2}-h^{2}.

$$\left(\frac{g}{g-h}-\frac{g^{2}}{\left(g+h\right)\left(g-h\right)}\right)\times \frac{4g+4h}{3g}$$

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

$$\left(\frac{g\left(g+h\right)}{\left(g+h\right)\left(g-h\right)}-\frac{g^{2}}{\left(g+h\right)\left(g-h\right)}\right)\times \frac{4g+4h}{3g}$$

Since \frac{g\left(g+h\right)}{\left(g+h\right)\left(g-h\right)} and \frac{g^{2}}{\left(g+h\right)\left(g-h\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{g\left(g+h\right)-g^{2}}{\left(g+h\right)\left(g-h\right)}\times \frac{4g+4h}{3g}$$

Do the multiplications in g\left(g+h\right)-g^{2}.

$$\frac{g^{2}+gh-g^{2}}{\left(g+h\right)\left(g-h\right)}\times \frac{4g+4h}{3g}$$

Combine like terms in g^{2}+gh-g^{2}.

$$\frac{gh}{\left(g+h\right)\left(g-h\right)}\times \frac{4g+4h}{3g}$$

Multiply \frac{gh}{\left(g+h\right)\left(g-h\right)} times \frac{4g+4h}{3g} by multiplying numerator times numerator and denominator times denominator.

$$\frac{gh\left(4g+4h\right)}{\left(g+h\right)\left(g-h\right)\times 3g}$$

Cancel out g in both numerator and denominator.

$$\frac{h\left(4g+4h\right)}{3\left(g+h\right)\left(g-h\right)}$$

Factor the expressions that are not already factored.

$$\frac{4h\left(g+h\right)}{3\left(g+h\right)\left(g-h\right)}$$

Cancel out g+h in both numerator and denominator.

$$\frac{4h}{3\left(g-h\right)}$$

Expand the expression.

$$\frac{4h}{3g-3h}$$