Solve a/6b^3+4/9b^2/frac{5{6b}-1/9b^3}

$$\frac { \frac { a } { 6 b ^ { 3 } } + \frac { 4 } { 9 b ^ { 2 } } } { \frac { 5 } { 6 b } - \frac { 1 } { 9 b ^ { 3 } } }$$

Evaluate

$$\frac{3a+8b}{15b^{2}-2}$$

$$b\neq 0\text{ and }|b|\neq \frac{\sqrt{30}}{15}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6b^{3} and 9b^{2} is 18b^{3}. Multiply \frac{a}{6b^{3}} times \frac{3}{3}. Multiply \frac{4}{9b^{2}} times \frac{2b}{2b}.

$$\frac{\frac{3a}{18b^{3}}+\frac{4\times 2b}{18b^{3}}}{\frac{5}{6b}-\frac{1}{9b^{3}}}$$

Since \frac{3a}{18b^{3}} and \frac{4\times 2b}{18b^{3}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3a+4\times 2b}{18b^{3}}}{\frac{5}{6b}-\frac{1}{9b^{3}}}$$

Do the multiplications in 3a+4\times 2b.

$$\frac{\frac{3a+8b}{18b^{3}}}{\frac{5}{6b}-\frac{1}{9b^{3}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6b and 9b^{3} is 18b^{3}. Multiply \frac{5}{6b} times \frac{3b^{2}}{3b^{2}}. Multiply \frac{1}{9b^{3}} times \frac{2}{2}.

$$\frac{\frac{3a+8b}{18b^{3}}}{\frac{5\times 3b^{2}}{18b^{3}}-\frac{2}{18b^{3}}}$$

Since \frac{5\times 3b^{2}}{18b^{3}} and \frac{2}{18b^{3}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3a+8b}{18b^{3}}}{\frac{5\times 3b^{2}-2}{18b^{3}}}$$

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

$$\frac{\frac{3a+8b}{18b^{3}}}{\frac{15b^{2}-2}{18b^{3}}}$$

Divide \frac{3a+8b}{18b^{3}} by \frac{15b^{2}-2}{18b^{3}} by multiplying \frac{3a+8b}{18b^{3}} by the reciprocal of \frac{15b^{2}-2}{18b^{3}}.

$$\frac{\left(3a+8b\right)\times 18b^{3}}{18b^{3}\left(15b^{2}-2\right)}$$

Cancel out 18b^{3} in both numerator and denominator.

$$\frac{3a+8b}{15b^{2}-2}$$

Expand

$$\frac{3a+8b}{15b^{2}-2}$$

$$b\neq 0\text{ and }|b|\neq \frac{\sqrt{30}}{15}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6b^{3} and 9b^{2} is 18b^{3}. Multiply \frac{a}{6b^{3}} times \frac{3}{3}. Multiply \frac{4}{9b^{2}} times \frac{2b}{2b}.

$$\frac{\frac{3a}{18b^{3}}+\frac{4\times 2b}{18b^{3}}}{\frac{5}{6b}-\frac{1}{9b^{3}}}$$

Since \frac{3a}{18b^{3}} and \frac{4\times 2b}{18b^{3}} have the same denominator, add them by adding their numerators.

$$\frac{\frac{3a+4\times 2b}{18b^{3}}}{\frac{5}{6b}-\frac{1}{9b^{3}}}$$

Do the multiplications in 3a+4\times 2b.

$$\frac{\frac{3a+8b}{18b^{3}}}{\frac{5}{6b}-\frac{1}{9b^{3}}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 6b and 9b^{3} is 18b^{3}. Multiply \frac{5}{6b} times \frac{3b^{2}}{3b^{2}}. Multiply \frac{1}{9b^{3}} times \frac{2}{2}.

$$\frac{\frac{3a+8b}{18b^{3}}}{\frac{5\times 3b^{2}}{18b^{3}}-\frac{2}{18b^{3}}}$$

Since \frac{5\times 3b^{2}}{18b^{3}} and \frac{2}{18b^{3}} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{3a+8b}{18b^{3}}}{\frac{5\times 3b^{2}-2}{18b^{3}}}$$

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

$$\frac{\frac{3a+8b}{18b^{3}}}{\frac{15b^{2}-2}{18b^{3}}}$$

Divide \frac{3a+8b}{18b^{3}} by \frac{15b^{2}-2}{18b^{3}} by multiplying \frac{3a+8b}{18b^{3}} by the reciprocal of \frac{15b^{2}-2}{18b^{3}}.

$$\frac{\left(3a+8b\right)\times 18b^{3}}{18b^{3}\left(15b^{2}-2\right)}$$

Cancel out 18b^{3} in both numerator and denominator.

$$\frac{3a+8b}{15b^{2}-2}$$