Đề bài

Rút gọn biểu thức: a) \(\sqrt {\frac{{{{13}^2} - {{12}^2}}}{{225}}} \) b) \(\frac{{\sqrt {{{\left( {6,2} \right)}^2} - {{\left( {5,9} \right)}^2}} }}{{\sqrt {2,43} }}\) c) \(\frac{{2 - \sqrt 2 }}{{\sqrt 2 }}\) d) \(\sqrt {6 + 2\sqrt 5 }  - 2\sqrt 5 \)

Lời giải chi tiết

a) \(\sqrt {\frac{{{{13}^2} - {{12}^2}}}{{225}}}  = \sqrt {\frac{{\left( {13 - 12} \right)\left( {13 + 12} \right)}}{{225}}} \) \(= \sqrt {\frac{{25}}{{225}}}  = \sqrt {\frac{1}{9}}  = \frac{1}{3}.\) b) \(\frac{{\sqrt {{{\left( {6,2} \right)}^2} - {{\left( {5,9} \right)}^2}} }}{{\sqrt {2,43} }} = \frac{{\sqrt {\left( {6,2 - 5,9} \right)\left( {6,2 + 5,9} \right)} }}{{\sqrt {2,43} }} \) \(= \frac{{\sqrt {0,3.12,1} }}{{\sqrt {2,43} }} = \sqrt {\frac{{3,63}}{{2,43}}}  = \sqrt {\frac{{121}}{{81}}}  = \frac{{11}}{9}.\) c) \(\frac{{2 - \sqrt 2 }}{{\sqrt 2 }} = \frac{{\sqrt 2 \left( {\sqrt 2  - 1} \right)}}{{\sqrt 2 }} = \sqrt 2  - 1.\) d) \(\sqrt {6 + 2\sqrt 5 }  - 2\sqrt 5  \) \(= \sqrt {1 + 2.1.\sqrt 5  + {{\left( {\sqrt 5 } \right)}^2}}  - 2\sqrt 5\\= \sqrt {{{\left( {1 + \sqrt 5 } \right)}^2}}  - 2\sqrt 5  \\= 1 + \sqrt 5  - 2\sqrt 5  = 1 - \sqrt 5 .\)