Basic Algebra combining exponent fractions/simplifying (George F Simmons "Precalculus in Nutshell")

Question

From George F. Simmons 'Precalculus' book, Algebra section, 5(d); Combine and Simplify

$$\frac{x}{xy^2} + \frac{y}{x^2y}$$

Combine: = $$\frac{x(x^2y) + y(xy^2)}{(xy^2)(x^2y)}$$ Simplify: = $$\frac{x(x^2y) + y(xy^2)}{xy(x^2y^2)}$$

The given answer, which I can't figure out how he arrives at it is: $$\frac{x^2 +y^2}{x^2y^2}$$

Answer 1

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

Answer 2

We have: $\dfrac{x}{xy^{2}}+\dfrac{y}{x^{2}y}$

$=\dfrac{x(x^{2}y)+y(xy^{2})}{(xy^{2})(x^{2}y)}$

$=\dfrac{x(x^{2}y)+y(xy^{2})}{xy(x^{2}y^{2})}$

This is where you got stuck, but there is not much left to do to arrive at that result.

$=\dfrac{x^{3}y+y^{3}x}{xy(x^{2}y^{2})}$

$=\dfrac{xy(x^{2}+y^{2})}{xy(x^{2}y^{2})}$

$=\dfrac{x^{2}+y^{2}}{x^{2}y^{2}}$