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Question
If `a/b = c/d` then prove that `(5a + 3b)/(5c + 3d) = ((4a^3 - 3b^3)/(4c^3 - 3d^3))^(1/3)`
Theorem
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Solution
`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.
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Chapter 7: Ratio and proportion - Exercise 7B [Page 125]
