Question

If `a/b = c/d` then prove that `(5a + 3b)/(5c + 3d) = ((4a^3 - 3b^3)/(4c^3 - 3d^3))^(1/3)`

Answer 1

`a/b = c/d` = k

a = bk

c = dk

L.H.S.

= `(5a + 3b)/(5c + 3d)`

= `(5bk + 3b)/(5dk + 3d)`

= `(b(5k + 3))/(d(5k + 3))`

= `b/d`

R.H.S.

= `((4a^3 - 3b^3)/(4c^3 - 3d^3))^(1/3)`

= `((4(bk)^3 - 3b^3)/(4(dk)^3 - 3d^3))^(1/3)`

= `((4b^3k^3 - 3b^3)/(4d^3k^3 - 3d^3))^(1/3)`

= `((b^3(4k^3 - 3))/(d^3(4k^3 - 3)))^(1/3)`

= `(b^3/d^3)^(1/3)`

= `((b/d)^3)^(1/3)`

= `b/d`

L.H.S. = R.H.S.