Question

Efetue :
[(a√a + b√b) . (√a + √b)^-1 + 3√ab]^1/2

A resposta é √a + √b . So que não consigo chegar

Answer 1

Faça em partes.
$\big(\sqrt{a}+\sqrt{b}\big)^{-1}= \frac{1}{\sqrt{a}+\sqrt{b}} =\frac{1}{\sqrt{a}+\sqrt{b}} \cdot\frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}-\sqrt{b}} =\frac{\sqrt{a}-\sqrt{b}}{(\sqrt{a})^2-(\sqrt{b})^2}=\frac{\sqrt{a}-\sqrt{b}}{a-b}\\ \\ ---------------------$
$\\\big(a\sqrt{a}+b\sqrt{b}\big)\cdot\big(\sqrt{a}+\sqrt{b}\big)^{-1}= \big(a\sqrt{a}+b\sqrt{b}\big)\cdot\frac{\sqrt{a}-\sqrt{b}}{a-b}\\ \\ =\frac{a\sqrt{a^2}-a\sqrt{ab}+b\sqrt{ab}-b\sqrt{b^2}}{a-b} \\ \\= \frac{a\cdot a-\sqrt{ab}\ (a-b)-b\cdot b}{a-b} = \frac{a^2-(a-b)\sqrt{ab}-b^2}{a-b} = \frac{a^2-b^2-(a-b)\sqrt{ab}}{a-b}\\ \\= \frac{(a-b)\cdot(a+b)-(a-b)\sqrt{ab}}{a-b}=\frac{(a-b)\cdot\big[(a+b)-\sqrt{ab}\big]}{a-b}=a+b-\sqrt{ab}$
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$\big[\big(a\sqrt{a} + b\sqrt{b}\big) \cdot \big(a\sqrt{a} + b\sqrt{b}\big)^{-1} + 3\sqrt{ab}\big]^{1/2}\\ \\ =\sqrt{\big(a\sqrt{a} + b\sqrt{b}\big) \cdot \big(a\sqrt{a} + b\sqrt{b}\big)^{-1} + 3\sqrt{ab}} \\ \\=\sqrt{a+b-\sqrt {ab} + 3\sqrt{ab}} \\ \\=\sqrt{a+b + 2\sqrt{ab}} \\ \\=\sqrt{a+ 2\sqrt{ab}+b} \\ \\=\sqrt{\big(\sqrt{a}+ \sqrt{b})^2} \\ \\=\sqrt{a}+ \sqrt{b}$