Упражнение 166

\({\largeа)}\ \frac{1-\frac{1}{x}}{1+\frac{1}{x}}=\frac{\left(1-\frac{1}{x}\right)\cdot{x}}{\left(1+\frac{1}{x}\right)\cdot{x}}=\frac{1\cdot{x}-\frac{1}{x}\cdot{x}}{1\cdot{x}+\frac{1}{x}\cdot{x}}=\frac{x-1}{x+1};\)

\({\largeб)}\ \frac{\frac{2a-b}{b}+1}{\frac{2a+b}{b}-1}=\frac{\left(\frac{2a-b}{b}+1\right)\cdot{b}}{\left(\frac{2a+b}{b}-1\right)\cdot{b}}=\frac{\frac{2a-b}{b}\cdot{b}+1\cdot{b}}{\frac{2a+b}{b}\cdot{b}-1\cdot{b}}=\frac{2a-b+b}{2a+b-b}=\frac{2a}{2a}=1;\)

\({\largeв)}\ \frac{\frac{x}{y^2}+\frac{y}{x^2}}{\frac{x}{y^2}-\frac{y}{x^2}}=\frac{\left(\frac{x}{y^2}+\frac{y}{x^2}\right)\cdot{x^2y^2}}{\left(\frac{x}{y^2}-\frac{y}{x^2}\right)\cdot{x^2y^2}}=\frac{\frac{x}{y^2}\cdot{x^2y^2}+\frac{y}{x^2}\cdot{x^2y^2}}{\frac{x}{y^2}\cdot{x^2y^2}-\frac{y}{x^2}\cdot{x^2y^2}}=\frac{x^3+y^3}{x^3-y^3};\)

\({\largeг)}\ \frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}{\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}}=\frac{\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)\cdot{abc}}{\left(\frac{1}{ab}+\frac{1}{bc}+\frac{1}{ac}\right)\cdot{abc}}=\frac{\frac{1}{a}\cdot{abc}+\frac{1}{b}\cdot{abc}+\frac{1}{c}\cdot{abc}}{\frac{1}{ab}\cdot{abc}+\frac{1}{bc}\cdot{abc}+\frac{1}{ac}\cdot{abc}}=\frac{bc+ac+ab}{c+a+b}=\frac{ab+ac+bc}{a+b+c}.\)