EDGE Math 8

Example 3

Simplify $\frac{2 x^{- 2} + 3 y^{- 2}}{x^{- 2} + y^{- 2}}$ and write the final answer without negative exponents.

Solution

\frac{2 x^{- 2} + 3 y^{- 2}}{x^{- 2} + y^{- 2}}

= \frac{\frac{2}{x^{2}} + \frac{3}{y^{2}}}{\frac{1}{x^{2}} + \frac{1}{y^{2}}}

= \frac{\frac{2 y^{2} + 3 x^{2}}{x^{2} y^{2}}}{\frac{y^{2} + x^{2}}{x^{2} y^{2}}}

= \frac{2 y^{2} + 3 x^{2}}{x^{2} y^{2}} \div \frac{y^{2} + x^{2}}{x^{2} y^{2}}

= \frac{2 y^{2} + 3 x^{2}}{x^{2} y^{2}} \cdot \frac{x^{2} y^{2}}{y^{2} + x^{2}}

= \frac{x^{2} y^{2} (2 y^{2} + 3 x^{2})}{x^{2} y^{2} (y^{2} + x^{2})}

= \frac{2 y^{2} + 3 x^{2}}{y^{2} + x^{2}}