Алгебра 8 класс учебник Макарычев, Миндюк ответы – номер 247

а) ab + ab/a + b(a + b/a − b − a − b)
1) a + b/a − b − a − b = a + b/a − b − a/1 − b/1 = a + b/a − b − a ⋅ (a − b)/1 ⋅ (a − b) − b ⋅ (a − b)/1 ⋅ (a − b) = a + b − a² + ab − ab + b²/a − b = a + b − a² + b²/a − b = (a + b) − (a − b)(a + b)/a − b = (a + b)(1 − a + b)/a − b
2) ab/a + b ⋅ (a + b)(1 − a + b)/a − b = ab ⋅ (a + b)(1 − a + b)/(a + b) ⋅ (a − b) = ab(a + b)(1 − a + b) : (a + b)/(a + b)(a − b) : (a + b) = ab(1 − a + b)/a − b
3) ab + ab(1 − a + b)/a − b = ab/1 + ab(1 − a + b)/a − b = ab ⋅ (a − b)/1 ⋅ (a − b) + ab(1 − a + b)/a − b = ab(a − b) + ab(1 − a + b)/a − b = ab(a − b + 1 − a + b)/a − b = ab ⋅ 1/a − b = ab/a − b
б) (y² − xy/x² + xy − xy + y²) ⋅ x/x − y + y/x + y
1) y² − xy/x² + xy − xy + y² = y² − xy/x² + xy − xy/1 + y²/1 = y² − xy/x² + xy − xy ⋅ (x² + xy)/1 ⋅ (x² + xy) + y² ⋅ (x² + xy)/1 ⋅ (x² + xy) = y² − xy − xy(x² + xy) + y²(x² + xy)/x² + xy = (y² − xy) + (x² + xy)(y² − xy)/x² + xy = (y² − xy)(1 + x² + xy)/x² + xy = y(y − y)(1 + x² + xy)/x(x + y)
2) y(y − y)(1 + x² + xy)/x(x + y) ⋅ x/x − y = −y(x − y)(1 + x² + xy) ⋅ x/x(x + y) ⋅ (x − y) = −y(x − y)(1 + x² + xy) ⋅ x : x(x − y)/x(x + y) ⋅ (x − y) : x(x − y) = −y(1 + x² + xy)/x + y
3) −y(1 + x² + xy)/x + y + y/x + y = −y − x²y − xy² + y/x + y = −x²y − xy²/x + y = −xy(x + y)/x + y = −xy(x + y) : (x + y)/(x + y) : (x + y) = −xy/1 = −xy
в) (1/(2a − b) + 2/4a² + 1/(2a + b)²) ⋅ 4a² + 4ab + b²/16a
1) 1/(2a − b) + 2/4a² + 1/(2a + b)² = 1/(2a − b)² + 2/(2a − b)(2a + b) + 1/(2a + b)² = 1 ⋅ (2a + b)²/(2a − b)² ⋅ (2a + b)² + 2 ⋅ (2a − b)(2a + b)/(2a − b)(2a + b) ⋅ (2a − b)(2a + b) + 1 ⋅ (2a − b)²/(2a + b)² ⋅ (2a − b)² = 4a² + 4ab + b² + 8a² − 2b² + 4a² − 4ab + b²/(2a − b)²(2a + b)² = 16a²/(2a − b)²(2a + b)²
2) 16a²/(2a − b)²(2a + b)² ⋅ 4a² + 4ab + b²/16a = 16a² ⋅ (2a + b)²/(2a − b)²(2a + b)² ⋅ 16a = 16a² ⋅ (2a + b)² : 16a(2a + b)²/(2a − b)²(2a + b)² ⋅ 16a : 16a(2a + b)² = a/(2a − b)²
г) 4c²/(c − 2)⁴ : (1/(c + 2)² + 1/(c − 2)² + 2/c² − 4)
1) 1/(c + 2)² + 1/(c − 2)² + 2/c² − 4 = 1/(c + 2)² + 1/(c − 2)² + 2/(c − 2)(c + 2) = 1 ⋅ (c − 2)²/(c + 2)² ⋅ (c − 2)² + 1 ⋅ (c + 2)²/(c − 2)² ⋅ (c + 2)² + 2 ⋅ (c − 2)(c + 2)/(c − 2)(c + 2) ⋅ (c − 2)(c + 2) = c² − 4c + 4 + c² + 4c + 4 + 2c² − 8/(c + 2)²(c − 2)² = 4c²/(c + 2)²(c − 2)²
2) 4c²/(c − 2)⁴ : 4c²/(c + 2)²(c − 2)² = 4c²/(c − 2)⁴ ⋅ (c + 2)²(c − 2)²/4c² = 4c² ⋅ (c + 2)²(c − 2)²/(c − 2)⁴ ⋅ 4c² = 4c² ⋅ (c + 2)²(c − 2)² : 4c²(c − 2)²/(c − 2)⁴ ⋅ 4c² : 4c²(c − 2)² = (c + 2)/(c − 2)²