Алгебра 8 класс учебник Макарычев, Миндюк ответы – номер 156

а) (1/y + 2/x − y)(x − x² + y²/x + y)
1) 1 ⋅ (x − y)/y ⋅ (x − y) + 2 ⋅ y/(x − y) ⋅ y = x − y + 2y/y(x − y) = x + y/y(x − y)
2) x − x² + y²/x + y = x/1 − x² + y²/x + y = x ⋅ (x + y)/1 ⋅ (x + y) − x² + y²/x + y = x(x + y) − x² − y²/(x + y) = x² + xy − x² − y²/(x + y) = xy − y²/(x + y) = y(x − y)/(x + y)
3) x + y/y(x − y) ⋅ y(x − y)/(x + y) = (x + y) ⋅ y(x − y)/y(x − y) ⋅ (x + y) = (x + y) ⋅ y(x − y) : y(x + y)(x − y)/y(x − y) ⋅ (x + y : y(x + y)(x − y)) = 1
б) (a + b − 2ab/a + b) : (a − b/a + b + b/a)
1) a + b − 2ab/a + b = a/1 + b/1 − 2ab/a + b = a ⋅ (a + b)/1 ⋅ (a + b) + b ⋅ (a + b)/1 ⋅ (a + b) − 2ab/a + b = a ⋅ (a + b) + b ⋅ (a + b) − 2ab/a + b = a² + ab + ab + b² − 2ab/a + b = a² + b²/a + b
2) a − b/a + b + b/a = (a − b) ⋅ a/(a + b) ⋅ a + b ⋅ (a + b)/a ⋅ (a + b) = (a − b) ⋅ a + b ⋅ (a + b)/a ⋅ (a + b) = a² − ab + ab + b²/a(a + b) = a² + b²/a(a + b)
3) a² + b²/a + b : a² + b²/a(a + b) = a² + b²/a + b ⋅ a(a + b)/a² + b² = (a² + b²) ⋅ a(a + b)/(a + b) ⋅ (a² + b²) = (a² + b²) ⋅ a(a + b) : (a + b)(a² + b²)/(a + b) ⋅ (a² + b²) : (a + b)(a² + b²) = a/1
в) (x² − 1)(1/x − 1 − 1/x + 1 + 1)
1) 1/x − 1 − 1/x + 1 = 1 ⋅ (x + 1)/(x − 1) ⋅ (x + 1) − 1 ⋅ (x − 1)/(x + 1) ⋅ (x − 1) = x + 1 − x + 1/(x − 1)(x + 1) = 2/(x − 1)(x + 1) = 2/x² − 1
2) 2/x² − 1 + 1 = 2/x² − 1 + x² − 1/x² − 1 = 2 + x² − 1/x² − 1 = x² + 1/x² − 1
3) (x² − 1) ⋅ x² + 1/x² − 1 = x² − 1/1 ⋅ x² + 1/x² − 1 = (x² − 1) ⋅ (x² + 1)/1 ⋅ (x² − 1) = (x² − 1) ⋅ (x² + 1) : (x² − 1)/1 ⋅ (x² − 1) : (x² − 1) = x² + 1/1 = x² + 1
г) (m + 1 − 1/1 − m) : (m − m²/m − 1)
1) m + 1 − 1/1 − m = m/1 + 1/1 − 1/1 − m = m ⋅ (1 − m)/1 ⋅ (1 − m) + 1 − m/1 − m − 1/1 − m = m − m² + 1 − m − 1/1 − m = −m²/1 − m
2) m − m²/m − 1 = m/1 − m²/m − 1 = m ⋅ (m − 1)/1 ⋅ (m − 1) − m²/m − 1 = m² − m − m²/m − 1 = −m/m − 1
3) −m²/1 − m : (−m/m − 1) = m²/m − 1 : (−m/m − 1) = m²/m − 1 ⋅ (m − 1/(−m)) = m² ⋅ (m − 1)/(m − 1) ⋅ (−m) = m² ⋅ (m − 1) : m(m − 1)/(m − 1) ⋅ (−m) : m(m − 1) = m/−1 = −m