Question

помогите пожалуйста ​

Answer 1

2. Выполните действия:

а) $( \frac{a}{ {b}^{2} } - \frac{1}{a} ) \div ( \frac{1}{b} - \frac{1}{a} ) = ( \frac{a \times a}{ {b}^{2} \times a} - \frac{1 \times {b}^{2} }{a \times {b}^{2} } ) \div ( \frac{1 \times a}{b \times a} - \frac{1 \times b}{a \times b} ) = ( \frac{ {a}^{2} }{a {b}^{2} } - \frac{ {b}^{2} }{a {b}^{2} } ) \div ( \frac{a}{ab} - \frac{b}{ab} ) = \frac{ {a}^{2} - {b}^{2} }{a {b}^{2} } \div \frac{a - b}{ab} = \frac{ {a}^{2} - {b}^{2} }{a {b}^{2} } \times \frac{ab}{a - b} = \frac{( {a}^{2} - {b}^{2} ) \times ab }{a {b}^{2} \times (a - b)} = \frac{(a - b)(a + b) \times ab}{ab \times b \times (a - b)} = \frac{a + b}{b}$

б) $( \frac{x}{5} + \frac{x}{8} ) \times \frac{1}{ {x}^{2} } = ( \frac{x \times 8}{5 \times 8} + \frac{x \times 5}{8 \times 5} ) \times \frac{1}{ {x}^{2} } = ( \frac{8x}{40} + \frac{5x}{40} ) \times \frac{1}{ {x}^{2} } = \frac{8x + 5x}{40} \times \frac{1}{ {x}^{2} } = \frac{13x}{40} \times \frac{1}{ {x}^{2} } = \frac{13x}{40 {x}^{2} } = \frac{13}{40x}$

в) $( \frac{x}{y} - \frac{y}{x} ) \div \frac{x - y}{3xy} = ( \frac{x \times x}{y \times x} - \frac{y \times y}{x \times y} ) \div \frac{x - y}{3xy} = ( \frac{ {x}^{2} }{xy} - \frac{ {y}^{2} }{xy} ) \div \frac{x - y}{3xy} = \frac{ {x}^{2} - {y}^{2} }{xy} \div \frac{x - y}{3xy} = \frac{ {x}^{2} - {y}^{2} }{xy} \times \frac{3xy}{x - y} = \frac{(x - y)(x + y) \times 3xy}{xy \times (x - y)} = \frac{(x + y) \times 3}{1} = 3(x + y) = 3x + 3y$

3. Выполните действия:

а) $(1 - \frac{1}{x} ) \div (1 + \frac{1}{x} ) = ( \frac{x}{x} - \frac{1}{x} ) \div ( \frac{x}{x} + \frac{1}{x} ) = \frac{x - 1}{x} \div \frac{x + 1}{x} = \frac{x - 1}{x} \times \frac{x}{x + 1} = \frac{(x - 1) \times x}{x \times (x + 1)} = \frac{x - 1}{x + 1}$

б) $(x - \frac{x}{y} ) \times (x + \frac{x}{y} ) = ( \frac{xy}{y} - \frac{x}{y} ) \times ( \frac{xy}{y} + \frac{x}{y} ) = \frac{xy - x}{y} \times \frac{xy + x}{y} = \frac{(xy - x) \times (xy + x)}{y \times y} = \frac{x(y - 1) \times x(y + 1)}{ {y}^{2} } = \frac{ {x}^{2} \times ( {y}^{2} - {1}^{2} )}{ {y}^{2} } = \frac{ {x}^{2} {y}^{2} - {x}^{2} }{ {y}^{2} }$

4. Упростите выражение:

а) ${(x + \frac{1}{x} )}^{2} - 2 = {( \frac{ {x}^{2} }{x} + \frac{ 1 }{x} )}^{2} - 2 = \frac{ {x}^{2} + 1 }{x} - 2 = \frac{ {x}^{2} + 1 }{x} - \frac{2x}{x} = \frac{ {x}^{2} + 1 - 2x }{x} = \frac{ {(x - 1)}^{2} }{x}$

б) ${(a + \frac{2}{a} )}^{2} - {(a - \frac{2}{a} )}^{2} = ((a + \frac{2}{a}) - (a - \frac{2}{a}) ) \times ((a + \frac{2}{a} ) + (a - \frac{2}{a} )) = (a + \frac{2}{a} - a + \frac{2}{a} ) \times (a + \frac{2}{a} + a - \frac{2}{a} ) = \frac{4}{a} \times 2a = \frac{8a}{a} = 8$

1. Упростить выражение:

а) $\frac{4}{ {b}^{2} - 4} + \frac{4b}{4 - 4b + {b}^{2} } \times ( \frac{2}{2b + {b}^{2} } - \frac{b}{4 + 2b} ) = \frac{4}{(b - 2)(b + 2)} + \frac{4b}{ {(2 - b)}^{2} } \times ( \frac{2}{b(2 + b)} - \frac{b}{2(2 + b)} ) = \frac{4}{(b - 2)(b + 2)} + \frac{4b}{ {(2 - b)}^{2} } \times \frac{4 - {b}^{2} }{2b(2 + b)} = \frac{4}{(b - 2)(b + 2)} + \frac{4b \times (2 - b)(2 + b)}{(2 - b)(2 - b) \times 2b(2 + b)} = \frac{4}{(b - 2)(b + 2)} + \frac{2 }{2 - b} = \frac{4}{(b - 2)(b + 2)} + \frac{ - 2}{b - 2} = \frac{4}{(b - 2)(b + 2)} + \frac{ - 2 \times (b + 2)}{(b - 2) \times (b + 2)} = \frac{4 - 2b - 4}{ {b}^{2} - 4 } = \frac{ - 2b}{ {b}^{2} - 4 } = \frac{2b}{4 - {b}^{2} }$

б) $( \frac{1 + x}{ {x}^{2} - xy} - \frac{1 - y}{ {y}^{2} - xy } ) \div \frac{ {x}^{2} + {y}^{2} + 2xy}{ {x}^{2} y - x {y}^{2} } - \frac{x}{ {x}^{2} - {y}^{2} } = ( \frac{1 + x}{x(x - y)} - \frac{1 - y}{y(y - x)} ) \div \frac{ {(x + y)}^{2} }{xy(x - y)} - \frac{x}{{x}^{2} - {y}^{2} } = ( \frac{1 + x}{x(x - y)} + \frac{1 - y}{y(x - y)} ) \div \frac{(x + y)(x + y)}{xy(x - y)} - \frac{x}{{x}^{2} - {y}^{2} } = ( \frac{y + xy}{xy(x - y)} + \frac{x - xy}{xy(x - y)} ) \div \frac{(x + y)(x + y)}{xy(x - y)} - \frac{x}{{x}^{2} - {y}^{2} } = \frac{y + xy + x - xy}{xy(x - y)} \times \frac{xy(x - y)}{(x + y)(x + y)} - \frac{x}{{x}^{2} - {y}^{2} } =$

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

в) на фото