Tag Archives: TrucCanThuc

Trục căn thức ở mẫu

Tính chất: Trục căn thức ở mẫu:

  • $\dfrac{1}{\sqrt A}=\dfrac {\sqrt A}{A}$.
  • $\dfrac {1}{\sqrt A-\sqrt B}=\dfrac {\sqrt A+\sqrt B}{A-B}$.
  • $\dfrac {1}{\sqrt A+\sqrt B}=\dfrac {\sqrt A-\sqrt B}{A-B}$.

Ví dụ: Trục căn thức ở mẫu:

a) $\dfrac {12\sqrt 2}{5\sqrt 3}$.

b) $\dfrac {3}{\sqrt 5-\sqrt 2}$.

c) $\dfrac {3+\sqrt 3}{1+\sqrt 2}+\dfrac {2+\sqrt 2}{2-\sqrt 2}$.

Giải

a) $\dfrac {12\sqrt 2}{5\sqrt 3}=\dfrac {12 \sqrt 2 \sqrt 3}{5.3}=\dfrac {4\sqrt 6}{5}$

b) $\dfrac {3}{\sqrt 5-\sqrt 2}=\dfrac{{3\left( {\sqrt 5 + \sqrt 2 } \right)}}{{\left( {\sqrt 5 – \sqrt 2 } \right)\left( {\sqrt 5 + \sqrt 2 } \right)}}=\dfrac {3\left (\sqrt 5+\sqrt 2 \right ) }{5-2}=\sqrt 5+\sqrt 2$

c) $\dfrac {3+\sqrt 3}{1+\sqrt 2}+\dfrac {2+\sqrt 2}{2-\sqrt 2}=\dfrac{{\left( {3 + \sqrt 3 } \right)\left( {\sqrt 2 – 1} \right)}}{{\left( {\sqrt 2 + 1} \right)\left( {\sqrt 2 – 1} \right)}} + \dfrac{{\left( {2 + \sqrt 2 } \right)\left( {2 + \sqrt 2 } \right)}}{{\left( {2 – \sqrt 2 } \right)\left( {2 + \sqrt 2 } \right)}}$

$=\dfrac{{3\sqrt 2 – 3 + \sqrt 6 – \sqrt 3 }}{{2 – 1}} + \dfrac{{6 + 4\sqrt 2 }}{{4 – 2}}$

$=3\sqrt 2 – 3 +\sqrt 6 – \sqrt 3 + 3 + 2\sqrt 2 =5\sqrt 2-\sqrt 3+\sqrt 6$

Bài tập:

Bài 1:  Trục căn thức ở mẫu:

a) $\dfrac{7}{\sqrt 3 }$; $\dfrac{3}{2\sqrt 5 }$; $\dfrac{5}{3\sqrt {12} }$; $\dfrac{2}{3\sqrt {20} }$.

b)$\dfrac{\sqrt 3 + 3}{5\sqrt 3 }$; $\dfrac{7 – \sqrt 7 }{\sqrt 7 – 1}$; $\dfrac{2}{\sqrt 5 + \sqrt 3 }$; $\dfrac{\sqrt 5 + 2}{\sqrt 5 – 2}$.

c) $\dfrac{y + a\sqrt y }{a\sqrt y }$; $\dfrac{b – \sqrt b }{\sqrt b – 1}$; $\dfrac{b}{5 + \sqrt b }$; $\dfrac{p}{2\sqrt p – 1}$.

Bài 2: Tính:

a) $\dfrac{1}{{2 – \sqrt 5 }} + \dfrac{1}{{2 + \sqrt 5 }}$.

b) $\dfrac{3}{2}\sqrt 6 + 2\sqrt {\dfrac{2}{3}} – 4\sqrt {\dfrac{3}{2}} $.

c) $\dfrac{{2 + \sqrt 3 }}{{2 – \sqrt 3 }} – \dfrac{{2 – \sqrt 3 }}{{2 + \sqrt 3 }}$.

d) $\dfrac{2}{{\sqrt 3 – 1}} + \dfrac{3}{{\sqrt 3 – 2}} + \dfrac{{12}}{{3 – \sqrt 3 }}$.

Bài 3: Rút gọn:

a) $\dfrac{{3 + 2\sqrt 3 }}{{\sqrt 3 }} – \dfrac{{2\sqrt 3 + \sqrt {15} }}{{\sqrt 5 + 2}}$.

b) $\dfrac{{5\sqrt 2 – 2\sqrt 5 }}{{\sqrt {10} }} – \dfrac{3}{{\sqrt 5 – \sqrt 2 }}$.

c) $\dfrac{{\sqrt {15} – \sqrt {12} }}{{\sqrt 5 – 2}} – \dfrac{1}{{2 – \sqrt 3 }}$.

d) $\dfrac{{\sqrt 2 }}{{\sqrt {\sqrt 2 + 1} – 1}} – \dfrac{{\sqrt 2 }}{{\sqrt {\sqrt 2 + 1} + 1}}$.