Вариант 1: 1. Сократите дробь: a) \(\frac{15xy^4}{10x^3y^2}\); б) \(\frac{ab-b}{b^2}\) 2. Выполните действия: a) \(\frac{a+b}{a-b} + \frac{a}{b}\); б) \(\frac{3x^2}{x^2-1} - \frac{3x}{x-1}\); в) \(\frac{a-3}{a} - \frac{2}{(a-3)^2} - \frac{a^2-9}{a}\)

Вариант 1

1. Сократите дробь:

1. a) \(\frac{15xy^4}{10x^3y^2} = \frac{3 \cdot 5 \cdot x \cdot y^2 \cdot y^2}{2 \cdot 5 \cdot x \cdot x^2 \cdot y^2} = \frac{3y^2}{2x^2}\)
2. б) \(\frac{ab-b}{b^2} = \frac{b(a-1)}{b^2} = \frac{a-1}{b}\)

2. Выполните действия:

1. a) \(\frac{a+b}{a-b} + \frac{a}{b} = \frac{b(a+b) + a(a-b)}{b(a-b)} = \frac{ab+b^2+a^2-ab}{b(a-b)} = \frac{a^2+b^2}{b(a-b)}\)
2. б) \(\frac{3x^2}{x^2-1} - \frac{3x}{x-1} = \frac{3x^2}{(x-1)(x+1)} - \frac{3x(x+1)}{(x-1)(x+1)} = \frac{3x^2 - 3x^2 - 3x}{(x-1)(x+1)} = \frac{-3x}{(x-1)(x+1)}\)
3. в) \(\frac{a-3}{a} - \frac{2}{(a-3)^2} - \frac{a^2-9}{a} = \frac{(a-3)^2(a-3) - 2a - (a^2-9)(a-3)}{a(a-3)^2} = \frac{(a-3)^3 - 2a - (a-3)(a+3)(a-3)}{a(a-3)^2}
   = \frac{(a-3)((a-3)^2 - 2 - (a+3)(a-3))}{a(a-3)^2} = \frac{(a-3)^2 - 2 - (a^2-9)}{a(a-3)} = \frac{a^2-6a+9 - 2 - a^2+9}{a(a-3)} = \frac{-6a+16}{a(a-3)}\)

Ответ: 1. а) \(\frac{3y^2}{2x^2}\); б) \(\frac{a-1}{b}\). 2. а) \(\frac{a^2+b^2}{b(a-b)}\); б) \(\frac{-3x}{(x-1)(x+1)}\); в) \(\frac{-6a+16}{a(a-3)}\).