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If x = a^((1)/(3)) b^((1)/(3)) + a^(-(1)...

If `x = a^((1)/(3)) b^((1)/(3)) + a^(-(1)/(3)) + a^(-(1)/(3)) b^((1)/(3))` then prove that `a(bx^(3) - 3bx - a) = b^(2)`

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`x = a^((1)/(3)) b^((1)/(3)) + a^(-(1)/(3)) + a^(-(1)/(3)) b^((1)/(3))`
`:. x^(3) = (a^((1)/(3)) b^(-(1)/(3)) + a^(-(1)/(3)) b^((1)/(3)))^(3)`
`= (a^((1)/(3)) b^(-(1)/(3)))^(3) + (a^(-(1)/(3)) b^((1)/(3)))^(3) + 3a^((1)/(3)) b^(-(1)/(3)).a^(-(1)/(3)) b^((1)/(3)) (a^((1)/(3)) b^(-(1)/(3)) + a^(-(1)/(3)) b^((1)/(3)))`
`= ab^(-1) + a^(-1) b + 3.a^(0).b^(0).x`
`= (a)/(b) + (b)/(a) + 3x = (a^(2) + b^(2) + 3abx)/(ab)`
`:. abx^(3) = a^(2) + b^(2) + 3abx`
or, `abx^(3) - 3abx = a^(2) + b^(2)`
or, `abx^(3) - 3abx - a^(2) = b^(2)`
or, `a (bx^(3) - 3bx - a) = b^(2)`
Hence, `a (bx^(3) - 3bx -a) = b^(2)`
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CALCUTTA BOOK HOUSE-QUADRATIC SURDS-Exercise -3.2
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