高専の数学解説

第1章 1.42

(1)

1x1+1x=1xx+1x=1x+1\dfrac{\dfrac{1}{x}}{1+\dfrac{1}{x}} = \dfrac{\dfrac{1}{x}}{\dfrac{x+1}{x}} = \dfrac{1}{x+1}

(2)

x1+1x−1=xx−1+1x−1=x−1\dfrac{x}{1+\dfrac{1}{x-1}} = \dfrac{x}{\dfrac{x-1+1}{x-1}} = x-1

(3)

x−2x+1xx+1−2=x(x+1)−2x+1x−2(x+1)x+1=x2+x−2x−2x−2=(x−1)(x+2)−x−2=1−x\dfrac{x-\dfrac{2}{x+1}}{\dfrac{x}{x+1}-2} = \dfrac{\dfrac{x(x+1)-2}{x+1}}{\dfrac{x-2(x+1)}{x+1}} = \dfrac{x^2+x-2}{x-2x-2} = \dfrac{(x-1)(x+2)}{-x-2} = 1-x

(4)

x+yyx−xy=x+yy2−x2xy=xy(x+y)(y−x)(y+x)=xyy−x\dfrac{x+y}{\dfrac{y}{x}-\dfrac{x}{y}} = \dfrac{x+y}{\dfrac{y^2-x^2}{xy}} = \dfrac{xy(x+y)}{(y-x)(y+x)} = \dfrac{xy}{y-x}

(5)

1+a−ba+b1−a−ba+b=a+b+a−ba+ba+b−a+ba+b=2a2b=ab\dfrac{1+\dfrac{a-b}{a+b}}{1-\dfrac{a-b}{a+b}} = \dfrac{\dfrac{a+b+a-b}{a+b}}{\dfrac{a+b-a+b}{a+b}} = \dfrac{2a}{2b} = \dfrac{a}{b}

解説: r31bn1z