Question

If x : a = y : b, prove that `(x^4 + a^4)/(x^3 + a^3) + (y^4 + b^4)/(y^3 + b^3) = ((x + y)^4 + (a + b)^4)/((x+ y)^3 + (a + b)^3`

Answer 1

`x/a = y/b` = k (say)
x = ak, y = bk
L.H.S. = `(x^4 + a^4)/(x^3 + a^3) + (y^4 + b^4)/(y^3 + b^3)`

= `(a^4k^4 + a^4)/(a^3k^3 + a^3) + (b^4k^4 + b^4)/(b^3k^3 + b^3)`

= `(a^4(k^4 + 1))/(a^3(k^3 + 1)) + (b^4(k^4 + 1))/(b^3(k^3 + 1)`

= `(a(k^4 + 1))/(k^3 + 1) + (b(k^4 + 1))/(k^3 + 1)`

= `(a(k^4 + 1) + b(k^4 + 1))/(k^3 + 1)`

= `((k^4 + 1)(a + b))/(k^3 + 1)`

R.H.S. = `((x + y)^4 + (a + b)^4)/((x+ y)^3 + (a + b)^3`

= `((ak + bk)^4 + (a + b)^4)/((ak + bk)^3 + (a + b)^3`

= `(k^4(a + b)^4 + (a - b)^4)/(k^3(a + b)^3(a + b)^3`

= `((a + b)^4(k^4 + 1))/((a + b)^3(k^3 + 1)`

= `((a + b)(k^4 + 1))/(k^3 + 1)`

= `((k^4 + 1)(a + b))/(k^3 + 1)`
∴ L.H.S. = R.H.S.
Hence proved.