Question

If a/b=c/d,show that:a³c⁺ac³/b³d⁺bd³=(a+c)⁴/(b+d)⁴

Answer 1

Given : a/b = c/d  

To find :   Show that   (a³c + ac³)/(b³d + bd³)  =  (a + c)⁴/(b + d)⁴

Solution:

Let say  a/b = c/d  = k

=> a = bk    & c  =  dk

(a³c + ac³)/(b³d + bd³)  =  (a + c)⁴/(b + d)⁴

LHS   =  (a³c + ac³)/(b³d + bd³)

= ( (bk)³(dk) + bk(dk)³) / ( bd(b² + d²))

=  ( (b³dk⁴ + bd³k⁴) / ( bd(b² + d²))

= k⁴bd(b² + d²) / ( bd(b² + d²))

= k⁴

RHS = (a + c)⁴/(b + d)⁴

= ( bk + dk)⁴/(b + d)⁴

= ((k(b + d))⁴ / (b + d)⁴

=  k⁴ (b + d)⁴ / (b + d)⁴

= k⁴

LHS = RHS = k⁴

=> (a³c + ac³)/(b³d + bd³)  =  (a + c)⁴/(b + d)⁴

QED

hence proved

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