Solve 1/beta^2+1/alpha^2

$$\frac { 1 } { \beta ^ { 2 } } + \frac { 1 } { \alpha ^ { 2 } }$$

Evaluate

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

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{1}{\beta ^{2}} times \frac{\alpha ^{2}}{\alpha ^{2}}. Multiply \frac{1}{\alpha ^{2}} times \frac{\beta ^{2}}{\beta ^{2}}.

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

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

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

Factor

$$\frac{\frac{\beta ^{2}}{\alpha ^{2}}+1}{\beta ^{2}}$$