881/VB02. problem / Math - Solved Math Problems

MR-881 / 13b. problem

Simplify the following expression:

$\frac{x^{- 4} - 25 y^{- 2}}{x^{- 2} - 5 y^{- 1}} \cdot x^{2} y {(y + 5 x^{2})}^{- 1}$

$\frac{x^{- 4} - 25 y^{- 2}}{x^{- 2} - 5 y^{- 1}} \cdot x^{2} y {(y + 5 x^{2})}^{- 1}$

$= \frac{(x^{- 2} - 5 y^{- 1}) \cdot (x^{- 2} + 5 y^{- 1})}{x^{- 2} - 5 y^{- 1}} \cdot x^{2} y \cdot \frac{1}{y + 5 x^{2}}$

$= (x^{- 2} + 5 y^{- 1}) \cdot x^{2} y \cdot \frac{1}{y + 5 x^{2}}$

$= (\frac{1}{x^{2}} + \frac{5}{1} \cdot \frac{1}{y}) \cdot x^{2} y \cdot \frac{1}{y + 5 x^{2}}$

$= (\frac{1}{x^{2}} + \frac{5}{y}) \cdot x^{2} y \cdot \frac{1}{y + 5 x^{2}}$

$= (\frac{1}{x^{2}} \cdot \frac{y}{y} + \frac{5}{y} \cdot \frac{x^{2}}{x^{2}}) \cdot x^{2} y \cdot \frac{1}{y + 5 x^{2}}$

$= (\frac{y}{x^{2} y} + \frac{5 x^{2}}{x^{2} y}) \cdot x^{2} y \cdot \frac{1}{y + 5 x^{2}}$

$= \frac{y + 5 x^{2}}{x^{2} y} \cdot x^{2} y \cdot \frac{1}{y + 5 x^{2}}$

$= 1$

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