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Question
If `"a"/"b" = "c"/"d"`, show that: `(a^3c + ac^3)/(b^3d + bd^3) = ("a" + "c")^4/("b" + "d")^4`
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Solution
Let `"a"/"b" = "c"/"d"` = k
⇒ a = bk and c = dk
L.H.S = `("a"^3"c" + "ac"^3)/("b"^3"d" + "bd"^3)`
= `(("bk")^3 xx ("dk")+("bk")("dk"^3))/("b"^3"d"+ "bd"^3)`
= `("b"^3"k"^3"dk"+"bkd"^3"k"^3)/("b"^3"d"+"bd"^3)`
= `("b"^3"dk"^4+"bd"^3"k"^4)/("b"^3"d"+"bd"^3)`
= `("k"^4("b"^3"d"+"bd"^3))/("b"^3"d"+"bd"^3)`
= k4
R.H.S = `("a" + "c")^4/("b" + "d")^4`
= `("bk"+"dk")^4/("b"+"d")^4`
= `("k"^4("b"+"d")^4)/("b"+"d")^4`
= k4
∴ L.H.S = R.H.S
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