Question

If `a/b = c/d`, show that: `(a^3c + ac^3)/(b^3d + bd^3) = (a + c)^4/(b + d)^4`.

Answer 1

Let `a/b = c/d = k`

`=>` a = bk and c = dk

L.H.S = `(a^3c + ac^3)/(b^3d + bd^3)`

= `(ac(a^2 + c^2))/(bd(b^2 + d^2))`

= `((bk xx dk)(b^2k^2 + d^2k^2))/(bd(b^2 + d^2))`

= `(k^2 xx k^2(b^2 + d^2))/(b^2 + d^2)`

= k⁴

R.H.S = `(a + c)^4/(b + d)^4`

= `(bk + dk)^4/(b + d)^4`

= `[(k(b + d))/(b + d)]^4`

= k⁴

Hence `(a^3c + ac^3)/(b^3d + bd^3) = (a + c)^4/(b + d)^4`