If A and B are the H.C.F. and L.C.M. respecti | vedclass

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(D) We know that for any two numbers or expressions $x$ and $y$,the product of their $H.C.F.$ $(A)$ and $L.C.M.$ $(B)$ is equal to the product of the

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$x^{3}+y^{3}$

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(D) We know that for any two numbers or expressions $x$ and $y$,the product of their $H.C.F.$ $(A)$ and $L.C.M.$ $(B)$ is equal to the product of the numbers themselves.So,$A 	imes B = x 	imes y$.We are given $A + B = x + y$.Consider the identity $(A+B)^3 = A^3 + B^3 + 3AB(A+B)$.Substituting the given values,we get $(x+y)^3 = A^3 + B^3 + 3xy(x+y)$.Rearranging for $A^3 + B^3$,we have $A^3 + B^3 = (x+y)^3 - 3xy(x+y)$.Factoring out $(x+y)$,we get $A^3 + B^3 = (x+y)((x+y)^2 - 3xy)$.Expanding the term inside the bracket: $A^3 + B^3 = (x+y)(x^2 + 2xy + y^2 - 3xy)$.$A^3 + B^3 = (x+y)(x^2 - xy + y^2)$.Using the algebraic identity $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$,we conclude that $A^3 + B^3 = x^3 + y^3$.

If $A$ and $B$ are the $H.C.F.$ and $L.C.M.$ respectively of two algebraic expressions $x$ and $y,$ and $A+B = x+y,$ then the value of $A^{3}+B^{3}$ is

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