Question

[Nguyễn Thị Yến Ngọc]

Cho a/b = c/d. Cm : 

a) (a-b/c-d) / ab/cd

b) (a+b/c+d)³=a³-b³/c³-d³

Answer 1

[Nguyễn Lê Phước Thịnh]

a: Đặt a/b=c/d=k

=>\(a=bk;c=dk\)

\(\left(\dfrac{a-b}{c-d}\right)^2=\left(\dfrac{bk-b}{dk-d}\right)^2=\left(\dfrac{b\left(k-1\right)}{d\left(k-1\right)}\right)^2=\left(\dfrac{b}{d}\right)^2\)

\(\dfrac{ab}{cd}=\dfrac{bk\cdot b}{dk\cdot k}=\dfrac{b^2\cdot k}{d^2\cdot k}=\dfrac{b^2}{d^2}\)

Do đó: \(\left(\dfrac{a-b}{c-d}\right)^2=\dfrac{ab}{cd}\)

b: \(\left(\dfrac{a+b}{c+d}\right)^3=\left(\dfrac{bk+b}{dk+d}\right)^3=\left(\dfrac{b\left(k+1\right)}{d\left(k+1\right)}\right)^3=\left(\dfrac{b}{d}\right)^3\)

\(\dfrac{a^3-b^3}{c^3-d^3}=\dfrac{b^3k^3-b^3}{d^3k^3-d^3}=\dfrac{b^3\left(k^3-1\right)}{d^3\left(k^3-1\right)}=\dfrac{b^3}{d^3}\)

Do đó: \(\left(\dfrac{a+b}{c+d}\right)^3=\dfrac{a^3-b^3}{c^3-d^3}\)