Алгебра 8 класс учебник Макарычев, Миндюк ответы – номер 166

а) $$ \frac{1-\frac{1}{x}}{1+\frac{1}{x}}=\frac{\frac{x}{x}-\frac{1}{x}}{\frac{x}{x}+\frac{1}{x}}=\frac{\frac{x-1}{x}}{\frac{x+1}{x}} $$ = x − 1/x : x + 1/x = x − 1/x ⋅ x/x + 1 = (x − 1) ⋅ x/x ⋅ (x + 1) = (x − 1) ⋅ x : x/x ⋅ (x + 1) : x = x − 1/x + 1
б) $$ \frac{\frac{2 a-b}{b}+1}{\frac{2 a+b}{b}-1}=\frac{\frac{2 a-b}{b}+\frac{b}{b}}{\frac{2 a+b}{b}-\frac{b}{b}}=\frac{\frac{2 a-b+b}{b}}{\frac{2 a+b-b}{b}}=\frac{\frac{2 a}{b}}{\frac{2 a}{b}} $$ = 2a/b : 2a/b = 2a/b ⋅ b/2a = 2a ⋅ b/b ⋅ 2a = 2a ⋅ b : 2ab/b ⋅ 2a : 2ab = 1
в) $$ \frac{\frac{x}{y^2}+\frac{y}{x^2}}{\frac{x}{y^2}-\frac{y}{x^2}}=\frac{\frac{x \cdot x^2}{y^2 \cdot x^2}+\frac{y \cdot y^2}{x^2 \cdot y^2}}{\frac{x \cdot x^2}{y^2 \cdot x^2}-\frac{y \cdot y^2}{x^2 \cdot y^2}}=\frac{\frac{x^3+y^3}{x^2 y^2}}{\frac{x^3-y^3}{x^2 y^2}} $$ = x³ + y³/x²y² : x³ − y³/x²y² = x³ + y³/x²y² ⋅ x²y²/x³ − y³ = (x³ + y³) ⋅ x²y²/x²y² ⋅ (x³ − y³) = (x³ + y³) ⋅ x²y² : x²y²/x²y² ⋅ (x³ − y³) : x²y² = x³ + y³/x³ − y³
г) $$ \frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}{\frac{1}{a b}+\frac{1}{b c}+\frac{1}{a c}}=\frac{\frac{1 \cdot b c}{a \cdot b c}+\frac{1 \cdot a c}{b \cdot a c}+\frac{1 \cdot a b}{c \cdot a b}}{\frac{1 \cdot c}{a b \cdot c}+\frac{1 \cdot a}{b c \cdot a}+\frac{1 \cdot b}{a c \cdot b}}=\frac{\frac{b c+a c+a b}{a b c}}{\frac{c+a+b}{a b c}} $$ = bc + ac + ab/abc : a + b + c/abc = bc + ac + ab/abc ⋅ abc/a + b + c = (bc + ac + ab) ⋅ abc/abc ⋅ (a + b + c) = (bc + ac + ab) ⋅ abc : abc/abc ⋅ (a + b + c) : abc = bc + ac + abc/a + b + c