Question

cho \(\dfrac{a}{b}=\dfrac{c}{d}\), chứng minh rằng:

a) \(\dfrac{3a^6+c^6}{3b^6+d^6}=\dfrac{\left(a+c\right)^6}{\left(b+d\right)^6}\)

b) \(\dfrac{a^2+b^2}{c^2+d^2}=\dfrac{5b-3a}{5c-3d}\)

Answer 1

ta có:

\(\dfrac{a}{b}=\dfrac{c}{d}\Rightarrow\dfrac{a^6}{b^6}=\dfrac{c^6}{d^6}=\dfrac{3a^6}{3b^6}\)

Áp dụng tính chất dãy tỉ sốbằng nhau ta có:

\(\dfrac{a^6}{b^6}=\dfrac{c^6}{d^6}=\dfrac{3a^6}{3b^6}=\dfrac{a^6+c^6}{b^6+d^6}=\dfrac{\left(a+c\right)^6}{\left(b+d\right)^6}\)

=\(\dfrac{c^6+3a^6}{d^6+3b^6}\)

\(\Rightarrow\dfrac{3a^6+c^6}{3b^6+d^6}=\dfrac{\left(a+c\right)^6}{\left(b+d\right)^6}\) (ĐPCM)