Đáp án:
\(\begin{array}{l}
a,\,\,\,6\sqrt 6 \\
b,\,\,\,2\sqrt 6 \\
c,\,\,\, - 2\sqrt 2 \\
2,\\
a,\,\,\,\sqrt {49 - 20\sqrt 6 } + \sqrt {25 + 4\sqrt 6 } = 6\\
b,\,\,\,\,2\sqrt 2 \left( {\sqrt 3 - 2} \right) + {\left( {1 + 2\sqrt 2 } \right)^2} - 2\sqrt 6 = 9
\end{array}\)
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
a,\,\,\,\sqrt {55 - 6\sqrt 6 } + \sqrt {55 + 6\sqrt 6 } \\
= \sqrt {54 - 6\sqrt 6 + 1} + \sqrt {54 + 6\sqrt 6 + 1} \\
= \sqrt {{{\left( {3\sqrt 6 } \right)}^2} - 2.3\sqrt 6 .1 + {1^2}} + \sqrt {{{\left( {3\sqrt 6 } \right)}^2} + 2.3\sqrt 6 .1 + {1^2}} \\
= \sqrt {{{\left( {3\sqrt 6 - 1} \right)}^2}} + \sqrt {{{\left( {3\sqrt 6 + 1} \right)}^2}} \\
= \left| {3\sqrt 6 - 1} \right| + \left| {3\sqrt 6 + 1} \right|\\
= \left( {3\sqrt 6 - 1} \right) + \left( {3\sqrt 6 + 1} \right)\\
= 6\sqrt 6 \\
b,\,\,\,\sqrt {15 + 6\sqrt 6 } - \sqrt {15 - 6\sqrt 6 } \\
= \sqrt {9 + 6\sqrt 6 + 6} - \sqrt {9 - 6\sqrt 6 + 6} \\
= \sqrt {{3^2} + 2.3.\sqrt 6 + {{\sqrt 6 }^2}} - \sqrt {{3^2} - 2.3.\sqrt 6 + {{\sqrt 6 }^2}} \\
= \sqrt {{{\left( {3 + \sqrt 6 } \right)}^2}} - \sqrt {{{\left( {3 - \sqrt 6 } \right)}^2}} \\
= \left| {3 + \sqrt 6 } \right| - \left| {3 - \sqrt 6 } \right|\\
= \left( {3 + \sqrt 6 } \right) - \left( {3 - \sqrt 6 } \right)\\
= 2\sqrt 6 \\
c,\,\,\,\sqrt {29 - 6\sqrt 6 } - \sqrt {29 + 6\sqrt 6 } \\
= \sqrt {27 - 6\sqrt 6 + 2} - \sqrt {27 + 6\sqrt 6 + 2} \\
= \sqrt {{{\left( {3\sqrt 3 } \right)}^2} - 2.3\sqrt 3 .\sqrt 2 + {{\sqrt 2 }^2}} - \sqrt {{{\left( {3\sqrt 3 } \right)}^2} + 2.3\sqrt 3 .\sqrt 2 + {{\sqrt 2 }^2}} \\
= \sqrt {{{\left( {3\sqrt 3 - \sqrt 2 } \right)}^2}} - \sqrt {{{\left( {3\sqrt 3 + \sqrt 2 } \right)}^2}} \\
= \left| {3\sqrt 3 - \sqrt 2 } \right| - \left| {3\sqrt 3 + \sqrt 2 } \right|\\
= \left( {3\sqrt 3 - \sqrt 2 } \right) - \left( {3\sqrt 3 + \sqrt 2 } \right)\\
= - 2\sqrt 2 \\
2,\\
a,\,\,\,\sqrt {49 - 20\sqrt 6 } + \sqrt {25 + 4\sqrt 6 } \\
= \sqrt {25 - 20\sqrt 6 + 24} + \sqrt {24 + 4\sqrt 6 + 1} \\
= \sqrt {{5^2} - 2.5.2\sqrt 6 + {{\left( {2\sqrt 6 } \right)}^2}} + \sqrt {{{\left( {2\sqrt 6 } \right)}^2} + 2.2\sqrt 6 .1 + {1^2}} \\
= \sqrt {{{\left( {5 - 2\sqrt 6 } \right)}^2}} + \sqrt {{{\left( {2\sqrt 6 + 1} \right)}^2}} \\
= \left| {5 - 2\sqrt 6 } \right| + \left| {2\sqrt 6 + 1} \right|\\
= 5 - 2\sqrt 6 + 2\sqrt 6 + 1\\
= 6\\
b,\,\,\,\,2\sqrt 2 \left( {\sqrt 3 - 2} \right) + {\left( {1 + 2\sqrt 2 } \right)^2} - 2\sqrt 6 \\
= 2\sqrt 2 .\sqrt 3 - 2\sqrt 2 .2 + \left[ {{1^2} + 2.1.2\sqrt 2 + {{\left( {2\sqrt 2 } \right)}^2}} \right] - 2\sqrt 6 \\
= 2\sqrt 6 - 4\sqrt 2 + 1 + 4\sqrt 2 + 8 - 2\sqrt 6 \\
= 9
\end{array}\)
