Question

If `a/b = c/d` then prove that each of the given ratio is equal to `root3 ((a^3 + 4c^3)/(b^3 + 4d^3))`.

Answer 1

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

Then,

a = bk

c = dk

`root3 ((a^3 + 4c^3)/(b^3 + 4d^3)) = root3 (((bk)^3 + 4(dk)^3)/(b^3 + 4d^3))`

= `root3 ((b^3k^3 + 4d^3k^3)/(b^3 + 4d^3))`

= `root3 ((k^3(b^3 + 4d^3))/(b^3 + 4d^3))`

= `root3 (k^3)`

= k

Since we established that the original ratios `a/b and c/d` both equal k, and the new ratio `root3 ((a^3 + 4c^3)/(b^3 + 4d^3))` also equals k they are all equal to each other:

`a/b = c/d = root3 ((a^3 + 4c^3)/(b^3 + 4d^3))`