Solve alpha/beta^2+beta/alpha^2

$$\frac { \alpha } { \beta ^ { 2 } } + \frac { \beta } { \alpha ^ { 2 } }$$

Evaluate

$$\frac{\alpha ^{3}+\beta ^{3}}{\left(\alpha \beta \right)^{2}}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of \beta ^{2} and \alpha ^{2} is \alpha ^{2}\beta ^{2}. Multiply \frac{\alpha }{\beta ^{2}} times \frac{\alpha ^{2}}{\alpha ^{2}}. Multiply \frac{\beta }{\alpha ^{2}} times \frac{\beta ^{2}}{\beta ^{2}}.

$$\frac{\alpha \alpha ^{2}}{\alpha ^{2}\beta ^{2}}+\frac{\beta \beta ^{2}}{\alpha ^{2}\beta ^{2}}$$

Since \frac{\alpha \alpha ^{2}}{\alpha ^{2}\beta ^{2}} and \frac{\beta \beta ^{2}}{\alpha ^{2}\beta ^{2}} have the same denominator, add them by adding their numerators.

$$\frac{\alpha \alpha ^{2}+\beta \beta ^{2}}{\alpha ^{2}\beta ^{2}}$$

Do the multiplications in \alpha \alpha ^{2}+\beta \beta ^{2}.

$$\frac{\alpha ^{3}+\beta ^{3}}{\alpha ^{2}\beta ^{2}}$$

Factor

$$\frac{\left(\alpha +\beta \right)\left(\alpha ^{2}-\alpha \beta +\beta ^{2}\right)}{\alpha ^{2}\beta ^{2}}$$