Question

Racionalize o denominador dos seguintes números irracionais:
a) 1
_____
√7

b) 2
_____
√5 - 1

c) √3 - 1
_____
√3 + 1​

Answer 1

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$\boxed{\begin{array}{l}\tt a)~\sf\dfrac{1}{\sqrt{7}}\cdot\dfrac{\sqrt{7}}{\sqrt{7}}=\dfrac{\sqrt{7}}{7}\\\tt b)~\sf\dfrac{2}{(\sqrt{5}-1)}\cdot\dfrac{(\sqrt{5}+1)}{(\sqrt{5}+1)}\\\sf=\dfrac{2\cdot(\sqrt{5}+1)}{(\sqrt{5})^2-1^2}\\\sf=\dfrac{2\cdot(\sqrt{5}+1)}{5-1}\\\sf=\dfrac{\diagdown\!\!\!2\cdot(\sqrt{5}+1)}{\diagdown\!\!\!\!4}=\dfrac{\sqrt{5}+1}{2}\\\tt \end{array}}$

$\large\boxed{\begin{array}{l}\tt c)~\sf\dfrac{(\sqrt{3}-1)}{(\sqrt{3}+1)}\cdot\dfrac{(\sqrt{3}-1)}{(\sqrt{3}-1)}\\\sf=\dfrac{3-2\sqrt{3}+1}{(\sqrt{3})^2-1^2}\\\sf=\dfrac{4-2\sqrt{3}}{3-1}=\dfrac{\diagdown\!\!\!\!2\cdot(2-\sqrt{3})}{\diagdown\!\!\!\!2}=2-\sqrt{3}\end{array}}$