a: \(\frac{25}{14x^2y}=\frac{25\cdot3\cdot y^4}{14x^2y\cdot3y^4}=\frac{75y^4}{42x^2y^5}\)
\(\frac{14}{21xy^5}=\frac{14\cdot2\cdot x}{21xy^5\cdot2x}=\frac{28x}{42x^2y^5}\)
b: \(\frac{11}{102x^4y}=\frac{11\cdot y^2}{102x^4y\cdot y^2}=\frac{11y^2}{102x^4y^3}\)
\(\frac{3}{34xy^3}=\frac{3\cdot3x^3}{34xy^3\cdot3x^3}=\frac{9x^3}{102x^4y^3}\)
c: \(\frac{3x+1}{12xy^4}=\frac{\left(3x+1\right)\cdot3\cdot x}{12xy^4\cdot3x}=\frac{9x^2+3x}{36x^2y^4}\)
\(\frac{y-2}{9x^2y^3}=\frac{\left(y-2\right)\cdot4\cdot y}{9x^2y^3\cdot4y}=\frac{4y^2-8y}{36x^2y^4}\)
d: \(\frac{1}{6x^3y^2}=\frac{1\cdot6\cdot y^2}{6x^3y^2\cdot6y^2}=\frac{6y^2}{36x^3y^4}\)
\(\frac{x+1}{9x^2y^4}=\frac{\left(x+1\right)\cdot4\cdot x}{9x^2y^4\cdot4x}=\frac{4x^2+4x}{36x^3y^4}\)
\(\frac{x-1}{4xy^3}=\frac{\left(x-1\right)\cdot9\cdot x^2}{4xy^3\cdot9x^2}=\frac{9x^3-9x^2}{36x^3y^3}=\frac{9x^3y-9x^2y}{36x^3y^4}\)
e: \(\frac{2x+3}{10x^4y}=\frac{\left(2x+3\right)\cdot12\cdot y^4}{10x^4y\cdot12y^4}=\frac{24xy^4+36y^4}{120x^4y^5}\)
\(\frac{5}{8x^2y^2}=\frac{5\cdot15\cdot x^2\cdot y^3}{8x^2y^2\cdot15x^2y^3}=\frac{75x^2y^3}{120x^4y^5}\)
\(\frac{2}{3xy^5}=\frac{2\cdot40\cdot x^3}{3xy^5\cdot40x^3}=\frac{80x^3}{120x^4y^5}\)
f: \(\frac{4x-4}{2x\left(x+3\right)}=\frac{4\left(x-1\right)}{2x\left(x+3\right)}=\frac{2\left(x-1\right)}{x\left(x+3\right)}=\frac{2\cdot\left(x-1\right)\cdot3\left(x+1\right)}{3x\left(x+3\right)\left(x+1\right)}=\frac{6x^2-6}{3x\cdot\left(x+3\right)\left(x+1\right)}\)
\(\frac{x-3}{3x\left(x+1\right)}=\frac{\left(x-3\right)\cdot\left(x+3\right)}{3x\left(x+1\right)\left(x+3\right)}=\frac{x^2-9}{3x\left(x+1\right)\left(x+3\right)}\)
g: \(\frac{2x}{\left(x+2\right)^3}=\frac{2x\cdot2x}{2x\cdot\left(x+2\right)^3}=\frac{4x^2}{2x\left(x+2\right)^3}\)
\(\frac{x-2}{2x\left(x+2\right)^2}=\frac{\left(x-2\right)\cdot\left(x+2\right)}{2x\left(x+2\right)^2\cdot\left(x+2\right)}=\frac{x^2-4}{2x\left(x+2\right)^3}\)
h: \(\frac{5}{3x^3-12x}=\frac{5}{3x\left(x^2-4\right)}=\frac{5}{3x\left(x-2\right)\left(x+2\right)}\)
\(=\frac{5\cdot2\cdot\left(x+3\right)}{3x\left(x-2\right)\left(x+2\right)\cdot2\left(x+3\right)}=\frac{10x+30}{6x\left(x-2\right)\left(x+2\right)\left(x+3\right)}\)
\(\frac{3}{\left(2x+4\right)\left(x+3\right)}=\frac{3}{2\left(x+2\right)\left(x+3\right)}=\frac{3\cdot3x\cdot\left(x-2\right)}{2\left(x+2\right)\left(x+3\right)\cdot3x\left(x-2\right)}=\frac{9x^2-18x}{6x\left(x+2\right)\left(x+3\right)\left(x-2\right)}\)
