[simplificação de expressoes] eliminar raizes

por **bira19** » Qui Out 06, 2011 23:33

$\frac{\left(5-{x}^{2} \right)3x\left(2x-4 \right)}{\left(\sqrt[2]{5}+x \right)\sqrt[2]{x-2}}$

Não consigo simplificar para eliminar raizes, como resolver?

por **LuizAquino** » Dom Out 09, 2011 09:26

bira19 escreveu: $\frac{\left(5-{x}^{2} \right)3x\left(2x-4 \right)}{\left(\sqrt{5}+x \right)\sqrt{x-2}}$

Não consigo simplificar para eliminar raizes, como resolver?

Siga o desenvolvimento:

$\frac{\left(5-{x}^{2} \right)3x\left(2x-4 \right)}{\left(\sqrt{5}+x \right)\sqrt{x-2}} = \frac{\left[\left(5-{x}^{2} \right)3x\left(2x-4 \right)\right]\cdot \left(\sqrt{5}-x \right)}{\left[\left(\sqrt{5}+x \right)\sqrt{x-2}\right]\cdot \left(\sqrt{5}-x \right)}$

$= \frac{\left(5-{x}^{2} \right)3x\left(2x-4 \right)\left(\sqrt{5}-x \right)}{\left(\sqrt{5}^2 - x^2 \right)\sqrt{x-2}}$

$= \frac{\cancel{\left(5-{x}^{2} \right)}3x\left(2x-4 \right)\left(\sqrt{5}-x \right)}{\cancel{\left(5-{x}^{2} \right)}\sqrt{x-2}}$

$= \frac{3x\left(2x-4 \right)\left(\sqrt{5}-x \right)}{\sqrt{x-2}}$

$= \frac{\left[3x\left(2x-4 \right)\left(\sqrt{5}-x \right)\right]\cdot \sqrt{x-2}}{\left(\sqrt{x-2}\right)\cdot \sqrt{x-2}}$

$= \frac{3x\left(2x-4 \right)\left(\sqrt{5}-x \right)\sqrt{x-2}}{x-2}$

$= \frac{3x\left[2(x-2)\right]\left(\sqrt{5}-x \right)\sqrt{x-2}}{x-2}$

$= \frac{6x(x-2)\left(\sqrt{5}-x \right)\sqrt{x-2}}{x-2}$

$= \frac{6x\cancel{(x-2)}\left(\sqrt{5}-x \right)\sqrt{x-2}}{\cancel{x-2}}$

$= 6x\left(\sqrt{5}-x \right)\sqrt{x-2}$