Question

Cho tỉ lệ thức \(\frac{a}{b}=\frac{c}{d}\). Chứng minh rằng \(\left(\frac{a-b}{c-d}\right)^4=\frac{a^4+b^4}{c^4+d^4}\)

 

Answer 1

dat :\(\frac{a}{b}=\frac{c}{d}=k\Rightarrow a=bk;b=dk\)

xet

\(\vec{ }\)\(\left(\frac{a-b}{c-d}\right)^4=\left(\frac{bk-b}{dk-d}\right)^4=\left(\frac{b\left(k-1\right)}{d\left(k-1\right)}\right)^4=\left(\frac{b}{d}\right)^4=\frac{b^4}{d^4}\left(1\right)\)

\(\frac{a^4+b^4}{c^4+d^4}=\frac{b^4k^4+b^4}{c^4k^4+d^4}=\frac{b^4\left(k^4+1\right)}{d^4\left(k^4+1\right)}=\frac{b^4}{d^4}\left(2\right)\)

tu(1) va (2)\(\Rightarrow\) \(\left(\frac{a-b}{c-d}\right)^4=\frac{a^4+b^4}{c^4+d^4}\left(DPCM\right)\)

Answer 2

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