Solve (4ab/6b^2+9-45b^2/9ab+10ab/2a^2)

$$( \frac { 4 a b } { 6 b ^ { 2 } } + \frac { 9 - 45 b ^ { 2 } } { 9 a b } + \frac { 10 a b } { 2 a ^ { 2 } } )$$

Evaluate

$$\frac{2a^{2}+3}{3ab}$$

Short Solution Steps

Cancel out 2b in both numerator and denominator.

$$\frac{2a}{3b}+\frac{9-45b^{2}}{9ab}+\frac{10ab}{2a^{2}}$$

Factor the expressions that are not already factored in \frac{9-45b^{2}}{9ab}.

$$\frac{2a}{3b}+\frac{9\left(-5b^{2}+1\right)}{9ab}+\frac{10ab}{2a^{2}}$$

Cancel out 9 in both numerator and denominator.

$$\frac{2a}{3b}+\frac{-5b^{2}+1}{ab}+\frac{10ab}{2a^{2}}$$

Cancel out 2a in both numerator and denominator.

$$\frac{2a}{3b}+\frac{-5b^{2}+1}{ab}+\frac{5b}{a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3b and ab is 3ab. Multiply \frac{2a}{3b} times \frac{a}{a}. Multiply \frac{-5b^{2}+1}{ab} times \frac{3}{3}.

$$\frac{2aa}{3ab}+\frac{3\left(-5b^{2}+1\right)}{3ab}+\frac{5b}{a}$$

Since \frac{2aa}{3ab} and \frac{3\left(-5b^{2}+1\right)}{3ab} have the same denominator, add them by adding their numerators.

$$\frac{2aa+3\left(-5b^{2}+1\right)}{3ab}+\frac{5b}{a}$$

Do the multiplications in 2aa+3\left(-5b^{2}+1\right).

$$\frac{2a^{2}-15b^{2}+3}{3ab}+\frac{5b}{a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3ab and a is 3ab. Multiply \frac{5b}{a} times \frac{3b}{3b}.

$$\frac{2a^{2}-15b^{2}+3}{3ab}+\frac{5b\times 3b}{3ab}$$

Since \frac{2a^{2}-15b^{2}+3}{3ab} and \frac{5b\times 3b}{3ab} have the same denominator, add them by adding their numerators.

$$\frac{2a^{2}-15b^{2}+3+5b\times 3b}{3ab}$$

Do the multiplications in 2a^{2}-15b^{2}+3+5b\times 3b.

$$\frac{2a^{2}-15b^{2}+3+15b^{2}}{3ab}$$

Combine like terms in 2a^{2}-15b^{2}+3+15b^{2}.

$$\frac{2a^{2}+3}{3ab}$$

Expand

$$\frac{2a^{2}+3}{3ab}$$

Short Solution Steps

Cancel out 2b in both numerator and denominator.

$$\frac{2a}{3b}+\frac{9-45b^{2}}{9ab}+\frac{10ab}{2a^{2}}$$

Factor the expressions that are not already factored in \frac{9-45b^{2}}{9ab}.

$$\frac{2a}{3b}+\frac{9\left(-5b^{2}+1\right)}{9ab}+\frac{10ab}{2a^{2}}$$

Cancel out 9 in both numerator and denominator.

$$\frac{2a}{3b}+\frac{-5b^{2}+1}{ab}+\frac{10ab}{2a^{2}}$$

Cancel out 2a in both numerator and denominator.

$$\frac{2a}{3b}+\frac{-5b^{2}+1}{ab}+\frac{5b}{a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3b and ab is 3ab. Multiply \frac{2a}{3b} times \frac{a}{a}. Multiply \frac{-5b^{2}+1}{ab} times \frac{3}{3}.

$$\frac{2aa}{3ab}+\frac{3\left(-5b^{2}+1\right)}{3ab}+\frac{5b}{a}$$

Since \frac{2aa}{3ab} and \frac{3\left(-5b^{2}+1\right)}{3ab} have the same denominator, add them by adding their numerators.

$$\frac{2aa+3\left(-5b^{2}+1\right)}{3ab}+\frac{5b}{a}$$

Do the multiplications in 2aa+3\left(-5b^{2}+1\right).

$$\frac{2a^{2}-15b^{2}+3}{3ab}+\frac{5b}{a}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 3ab and a is 3ab. Multiply \frac{5b}{a} times \frac{3b}{3b}.

$$\frac{2a^{2}-15b^{2}+3}{3ab}+\frac{5b\times 3b}{3ab}$$

Since \frac{2a^{2}-15b^{2}+3}{3ab} and \frac{5b\times 3b}{3ab} have the same denominator, add them by adding their numerators.

$$\frac{2a^{2}-15b^{2}+3+5b\times 3b}{3ab}$$

Do the multiplications in 2a^{2}-15b^{2}+3+5b\times 3b.

$$\frac{2a^{2}-15b^{2}+3+15b^{2}}{3ab}$$

Combine like terms in 2a^{2}-15b^{2}+3+15b^{2}.

$$\frac{2a^{2}+3}{3ab}$$