Ta có:
$\dfrac{a}{b} = \dfrac{c}{d}$
$⇒ (\dfrac{a}{b})^{6} = (\dfrac{c}{d})^{6}$
$⇒ \dfrac{a^{6}}{b^{6}} = \dfrac{c^{6}}{d^{6}}$
$⇒ \dfrac{3a^{6}}{3b^{6}} = \dfrac{3c^{6}}{3d^{6}}$
Áp dụng tính chất dãy tỉ số bằng nhau, ta có:
$\dfrac{3a^{6}}{3b^{6}} = \dfrac{3c^{6}}{3d^{6}} = \dfrac{3a^{6} + 3c^{6}}{3b^{6} + 3d^{6}} = \dfrac{3(a + c)^{6}}{3(b + d)^{6}} = \dfrac{(a + c)^{6}}{(b + d)^{6}} (đpcm)$