Simplify the following expressions to their simplest form:
a) $\frac{a b\left(x^{2}+y^{2}\right)+x y\left(a^{2}+b^{2}\right)}{a b\left(x^{2}-y^{2}\right)+x y\left(a^{2}-b^{2}\right)}$
b) $\frac{\left(a^{n+x}-a^{n}\right)\left(a^{n}-a^{n-x}\right)}{\left(a^{n+x}-a^{n}\right)-\left(a^{n}-a^{n-x}\right)}$
a) Removing the parentheses in the numerator and denominator

$$
\frac{a b x^{2}+a b y^{2}+a^{2} x y+b^{2} x y}{a b x^{2}-a b y^{2}+a^{2} x y-b^{2} x y}
$$

Factoring out $ax$ from the odd-numbered terms and $by$ from the even-numbered terms in the numerator, and $-by$ from the even-numbered terms in the denominator

$$
\frac{a x(b x+a y)+b y(a y+b y)}{a x(b x+a y)-b y(a y+b y)}=\frac{(b x+a y)(a x+b y)}{(b x+a y)(a x-b y)}=\frac{a x+b y}{a x-b y}
$$

b) Factoring out $a$ from the first factor in the numerator:

$$
\frac{a^{n}\left(a^{x}-1\right)\left(a^{n}-a^{n-x}\right)}{\left(a^{n+x}-a^{n}\right)-\left(a^{n}-a^{n-x}\right)}=\frac{a^{n}\left(a^{n+x}-a^{n}-a^{n}+a^{n-x}\right)}{a^{n+x}-a^{n}-a^{n}+a^{n-x}}=a^{n}
$$

Csomós Sándor (Hatvan, 4th grade chemical engineering technician I. o. t.)