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Math Help - Inequality rational power

  1. #1
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    Inequality rational power

    Show that if a,b and c d are positive ( and c and d are rational), then (a^c-b^c)(a^d-b^d) \geq 0 Can we proceed by multiplying the factors like :

    a^{c+d}-b^ca^d-a^cb^d+b^{c+d} \geq 0 Can somebody brief me the concept behind this..... I will be greatful....
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  2. #2
    Junior Member Nehushtan's Avatar
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    Re: Inequality rational power

    Given a,b\in\mathbb R, either a\geqslant b or a\leqslant b. Moreover, since a,b,c,d are positive, a\geqslant b \implies a^c\geqslant b^c and a^d\geqslant b^d; similarly for a\leqslant b. In both cases, a^c-b^c and a^d-b^d have the same sign (or are zero) and so their product is non-negative.
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