Thread: Inequality rational power

1. 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....

2. 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.