Giải Bài 4 trang 39 SGK Toán 8 tập 1 – Chân trời sáng tạo

Đề bài

Tính:

a) \(\left( {\dfrac{{1 – x}}{x} + {x^2} – 1} \right):\dfrac{{x – 1}}{x}\)       

b) \(\left( {\dfrac{1}{{{x^2}}} – \dfrac{1}{x}} \right) \cdot \dfrac{{{x^2}}}{y} + \dfrac{x}{y}\)                

c) \(\dfrac{3}{x} – \dfrac{2}{x}:\dfrac{1}{x} + \dfrac{1}{x} \cdot \dfrac{{{x^2}}}{3}\)

Lời giải chi tiết

a)

\(\left( {\dfrac{{1 – x}}{x} + {x^2} – 1} \right):\dfrac{{x – 1}}{x}\) \( \\= \left( { – \dfrac{{x – 1}}{x} + \dfrac{{\left( {{x^2} – 1} \right)x}}{x}} \right) \cdot \dfrac{x}{{x – 1}} \\= \dfrac{{ – \left( {x – 1} \right) + x\left( {x – 1} \right)\left( {x + 1} \right)}}{x} \cdot \dfrac{x}{{x – 1}} \\= \dfrac{{\left( {x – 1} \right)\left[ { x\left( {x + 1}  \right) – 1} \right]}}{x} \cdot \dfrac{x}{{x – 1}}\)

\( = x\left( {x + 1} \right) – 1 = {x^2} + x -1\)  

b)

\(\left( {\dfrac{1}{{{x^2}}} – \dfrac{1}{x}} \right) \cdot \dfrac{{{x^2}}}{y} + \dfrac{x}{y}\)

\(\begin{array}{l} = \left( {\dfrac{1}{{{x^2}}} – \dfrac{x}{{{x^2}}}} \right) \cdot \dfrac{{{x^2}}}{y} + \dfrac{x}{y}\\ = \dfrac{{1 – x}}{{{x^2}}} \cdot \dfrac{{{x^2}}}{y} + \dfrac{x}{y}\\ = \dfrac{{1 – x}}{y} + \dfrac{x}{y}\\ = \dfrac{1}{y}\end{array}\)

c)

\(\dfrac{3}{x} – \dfrac{2}{x}:\dfrac{1}{x} + \dfrac{1}{x} \cdot \dfrac{{{x^2}}}{3}\)

\(\begin{array}{l} = \dfrac{3}{x} – \dfrac{2}{x} \cdot \dfrac{x}{1} + \dfrac{1}{x} \cdot \dfrac{{{x^2}}}{3}\\ = \dfrac{3}{x} – 2 + \dfrac{x}{3}\\ = \dfrac{9}{{3x}} – \dfrac{{6x}}{{3x}} + \dfrac{{{x^2}}}{{3x}}\\ = \dfrac{{{x^2} – 6x + 9}}{{3x}}\end{array}\)