Solve 1/2a^3+1/4ab^2+2/8a^2b^2

$$\frac { 1 } { 2 a ^ { 3 } } + \frac { 1 } { 4 a b ^ { 2 } } + \frac { 2 } { 8 a ^ { 2 } b ^ { 2 } }$$

Evaluate

$$\frac{a^{2}+a+2b^{2}}{4b^{2}a^{3}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 2a^{3} and 4ab^{2} is 4b^{2}a^{3}. Multiply \frac{1}{2a^{3}} times \frac{2b^{2}}{2b^{2}}. Multiply \frac{1}{4ab^{2}} times \frac{a^{2}}{a^{2}}.

$$\frac{2b^{2}}{4b^{2}a^{3}}+\frac{a^{2}}{4b^{2}a^{3}}+\frac{2}{8a^{2}b^{2}}$$

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

$$\frac{2b^{2}+a^{2}}{4b^{2}a^{3}}+\frac{2}{8a^{2}b^{2}}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4b^{2}a^{3} and 8a^{2}b^{2} is 8b^{2}a^{3}. Multiply \frac{2b^{2}+a^{2}}{4b^{2}a^{3}} times \frac{2}{2}. Multiply \frac{2}{8a^{2}b^{2}} times \frac{a}{a}.

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

Since \frac{2\left(2b^{2}+a^{2}\right)}{8b^{2}a^{3}} and \frac{2a}{8b^{2}a^{3}} have the same denominator, add them by adding their numerators.

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

Do the multiplications in 2\left(2b^{2}+a^{2}\right)+2a.

$$\frac{4b^{2}+2a^{2}+2a}{8b^{2}a^{3}}$$

Factor the expressions that are not already factored in \frac{4b^{2}+2a^{2}+2a}{8b^{2}a^{3}}.

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

Cancel out 2 in both numerator and denominator.

$$\frac{a^{2}+a+2b^{2}}{4b^{2}a^{3}}$$

Factor

$$\frac{a^{2}+a+2b^{2}}{4b^{2}a^{3}}$$