Question

Упростите выражение:

Answer 1

$(1+ \frac{1}{x+\frac{1}{x+\frac{1}{x-1}}})\cdot\frac{x^3-(x-1)^2}{x^2+2}= (1+ \frac{1}{x+\frac{1}{\frac{x^2-x+1}{x-1}}})\cdot\frac{x^3-(x-1)^2}{x^2+2}= \\\ =(1+ \frac{1}{x+\frac{x-1}{x^2-x+1}})\cdot\frac{x^3-(x-1)^2}{x^2+2}= (1+ \frac{1}{\frac{x^3-x^2+x+x-1}{x^2-x+1}})\cdot\frac{x^3-(x-1)^2}{x^2+2}= \\\ =(1+ \frac{x^2-x+1}{x^3-x^2+x+x-1})\cdot\frac{x^3-(x-1)^2}{x^2+2}= ( \frac{x^3-x^2+x+x-1+x^2-x+1}{x^3-x^2+x+x-1})\cdot\frac{x^3-(x-1)^2}{x^2+2}=$$= \frac{x^3+x}{x^3-x^2+2x-1}\cdot\frac{x^3-x^2+2x-1}{x^2+2}=\frac{x^3+x}{x^2+2}$