Question

racionalize o denominador de cada uma das seguintes frações:
$\frac{2}{ \sqrt{3} }$
b)
$\frac{1}{ \sqrt{2} }$
c)
$\frac{ \sqrt{3} }{ \sqrt{5} }$
d)
$\frac{1}{2 + \sqrt{2} }$
ajuda prvf!!!
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Answer 1

$\textrm{a)}\ \dfrac{2}{\sqrt3}=\dfrac{2}{\sqrt3}\times 1=\dfrac{2}{\sqrt3}\times\dfrac{\sqrt3}{\sqrt3}=\dfrac{2\times\sqrt3}{\sqrt3\times\sqrt3}=\dfrac{2\sqrt3}{\sqrt{3\times3}}=\dfrac{2\sqrt3}{\sqrt{3^2}}=\dfrac{2\sqrt3}{\sqrt3^2}=\dfrac{2\sqrt3}{3}$

$\textrm{b)}\ \dfrac{1}{\sqrt2}=\dfrac{1}{\sqrt2}\times 1=\dfrac{1}{\sqrt2}\times\dfrac{\sqrt2}{\sqrt2}=\dfrac{1\times\sqrt2}{\sqrt2\times\sqrt2}=\dfrac{1\sqrt2}{\sqrt{2\times2}}=\dfrac{\sqrt2}{\sqrt{2^2}}=\dfrac{\sqrt2}{\sqrt{2}^2}=\dfrac{\sqrt2}2$

$\textrm{c)}\ \dfrac{\sqrt3}{\sqrt5}=\dfrac{\sqrt3}{\sqrt5}\times 1=\dfrac{\sqrt3}{\sqrt5}\times\dfrac{\sqrt5}{\sqrt5}=\dfrac{\sqrt3\times\sqrt5}{\sqrt5\times\sqrt5}=\dfrac{\sqrt{3\times5}}{\sqrt{5\times5}}=\dfrac{\sqrt{15}}{\sqrt{5^2}}=\dfrac{\sqrt{15}}{\sqrt{5}^2}=\dfrac{\sqrt{15}}{5}$

$\textrm{d)}\ \dfrac{1}{2+\sqrt2}\\\\\\=\dfrac{1}{2+\sqrt2}\times 1\\\\\\=\dfrac{1}{2+\sqrt2}\times\dfrac{{2-\sqrt2}}{{2-\sqrt2}}\\\\\\=\dfrac{1\times(2-\sqrt2)}{(2+\sqrt2)\times(2-\sqrt2)}\\\\\\=\dfrac{2-\sqrt2}{2^2-\sqrt2^2}\\\\\\=\dfrac{2-\sqrt2}{4-2}\\\\\\=\dfrac{2-\sqrt2}{2}$