Đáp án:
\(\begin{array}{l}
a,\,\,\,\,30\\
b,\,\,\,\,\sqrt 3 - 9\sqrt 2 \\
c,\,\,\,\,1\\
d,\,\,\,\,6\\
e,\,\,\,\, - 2\sqrt 3 \\
f,\,\,\,1\\
g,\,\,\,\,4\\
h,\,\,\,\,\dfrac{{ - 4\sqrt 3 }}{{13}}\\
i,\,\,\,\,1\\
k,\,\,\,\,36\\
l,\,\,\,\,4\sqrt 3 - 2\\
m,\,\,\,2
\end{array}\)
Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
a,\\
\left( {3\sqrt 8 - \sqrt {18} + 2\sqrt {72} } \right).\sqrt 2 \\
= \left( {3\sqrt {{2^2}.2} - \sqrt {{3^2}.2} + 2\sqrt {{6^2}.2} } \right).\sqrt 2 \\
= \left( {3.2\sqrt 2 - 3\sqrt 2 + 2.6\sqrt 2 } \right).\sqrt 2 \\
= \left( {6\sqrt 2 - 3\sqrt 2 + 12\sqrt 2 } \right).\sqrt 2 \\
= 15\sqrt 2 .\sqrt 2 \\
= 15.2\\
= 30\\
b,\\
2\sqrt {48} - 2\sqrt {98} + \sqrt {50} - \sqrt {147} \\
= 2\sqrt {16.3} - 2\sqrt {49.2} + \sqrt {25.2} - \sqrt {49.3} \\
= 2\sqrt {{4^2}.3} - 2\sqrt {{7^2}.2} + \sqrt {{5^2}.2} - \sqrt {{7^2}.3} \\
= 2.4\sqrt 3 - 2.7\sqrt 2 + 5\sqrt 2 - 7\sqrt 3 \\
= 8\sqrt 3 - 14\sqrt 2 + 5\sqrt 2 - 7\sqrt 3 \\
= \sqrt 3 - 9\sqrt 2 \\
c,\\
\sqrt {4 - 2\sqrt 3 } + \sqrt {7 - 4\sqrt 3 } \\
= \sqrt {3 - 2.\sqrt 3 .1 + 1} + \sqrt {4 - 2.2\sqrt 3 + 3} \\
= \sqrt {{{\sqrt 3 }^2} - 2.\sqrt 3 .1 + {1^2}} + \sqrt {{2^2} - 2.2.\sqrt 3 + {{\sqrt 3 }^2}} \\
= \sqrt {{{\left( {\sqrt 3 - 1} \right)}^2}} + \sqrt {{{\left( {2 - \sqrt 3 } \right)}^2}} \\
= \left| {\sqrt 3 - 1} \right| + \left( {2 - \sqrt 3 } \right)\\
= \left( {\sqrt 3 - 1} \right) + \left( {2 - \sqrt 3 } \right)\\
= 1\\
d,\\
\sqrt {12 + 6\sqrt 3 } + \sqrt {12 - 6\sqrt 3 } \\
= \sqrt {9 + 6\sqrt 3 + 3} + \sqrt {9 - 6\sqrt 3 + 3} \\
= \sqrt {{3^2} + 2.3.\sqrt 3 + 3} + \sqrt {{3^2} - 2.3.\sqrt 3 + 3} \\
= \sqrt {{{\left( {3 + \sqrt 3 } \right)}^2}} + \sqrt {{{\left( {3 - \sqrt 3 } \right)}^2}} \\
= \left| {3 + \sqrt 3 } \right| + \left| {3 - \sqrt 3 } \right|\\
= \left( {3 + \sqrt 3 } \right) + \left( {3 - \sqrt 3 } \right)\\
= 6\\
e,\\
\sqrt {8 - 2\sqrt {15} } - \sqrt {8 + 2\sqrt {15} } \\
= \sqrt {5 - 2\sqrt {15} + 3} - \sqrt {5 + 2\sqrt {15} + 3} \\
= \sqrt {{{\sqrt 5 }^2} - 2.\sqrt 5 .\sqrt 3 + {{\sqrt 3 }^2}} - \sqrt {{{\sqrt 5 }^2} + 2.\sqrt 5 .\sqrt 3 + {{\sqrt 3 }^2}} \\
= \sqrt {{{\left( {\sqrt 5 - \sqrt 3 } \right)}^2}} - \sqrt {{{\left( {\sqrt 5 + \sqrt 3 } \right)}^2}} \\
= \left| {\sqrt 5 - \sqrt 3 } \right| - \left| {\sqrt 5 + \sqrt 3 } \right|\\
= \left( {\sqrt 5 - \sqrt 3 } \right) - \left( {\sqrt 5 + \sqrt 3 } \right)\\
= - 2\sqrt 3 \\
f,\\
\sqrt {{{\left( {\sqrt 5 - 3} \right)}^2}} + \sqrt {{{\left( {2 - \sqrt 5 } \right)}^2}} \\
= \left| {\sqrt 5 - 3} \right| + \left| {2 - \sqrt 5 } \right|\\
= \left( {3 - \sqrt 5 } \right) + \left( {\sqrt 5 - 2} \right)\\
= 1\\
g,\\
\dfrac{3}{{\sqrt 7 - 2}} + \dfrac{{ - 3}}{{\sqrt 7 + 2}}\\
= \dfrac{{3.\left( {\sqrt 7 + 2} \right) - 3.\left( {\sqrt 7 - 2} \right)}}{{\left( {\sqrt 7 - 2} \right)\left( {\sqrt 7 + 2} \right)}}\\
= \dfrac{{3\sqrt 7 + 6 - 3\sqrt 7 + 6}}{{{{\sqrt 7 }^2} - {2^2}}}\\
= \dfrac{{12}}{{7 - 4}}\\
= 4\\
h,\\
\dfrac{1}{{5 + 2\sqrt 3 }} - \dfrac{1}{{5 - 2\sqrt 3 }}\\
= \dfrac{{\left( {5 - 2\sqrt 3 } \right) - \left( {5 + 2\sqrt 3 } \right)}}{{\left( {5 + 2\sqrt 3 } \right)\left( {5 - 2\sqrt 3 } \right)}}\\
= \dfrac{{ - 4\sqrt 3 }}{{{5^2} - {{\left( {2\sqrt 3 } \right)}^2}}}\\
= \dfrac{{ - 4\sqrt 3 }}{{25 - 12}}\\
= \dfrac{{ - 4\sqrt 3 }}{{13}}\\
i,\\
\dfrac{{5\sqrt 2 - 2\sqrt 5 }}{{\sqrt 5 - \sqrt 2 }} - \dfrac{9}{{\sqrt {10} + 1}}\\
= \dfrac{{{{\sqrt 5 }^2}.\sqrt 2 - {{\sqrt 2 }^2}.\sqrt 5 }}{{\sqrt 5 - \sqrt 2 }} - \dfrac{{9\left( {\sqrt {10} - 1} \right)}}{{\left( {\sqrt {10} - 1} \right)\left( {\sqrt {10} + 1} \right)}}\\
= \dfrac{{\sqrt 5 .\sqrt 2 .\left( {\sqrt 5 - \sqrt 2 } \right)}}{{\sqrt 5 - \sqrt 2 }} - \dfrac{{9\left( {\sqrt {10} - 1} \right)}}{{{{\sqrt {10} }^2} - {1^2}}}\\
= \sqrt {10} - \dfrac{{9\left( {\sqrt {10} - 1} \right)}}{9}\\
= \sqrt {10} - \left( {\sqrt {10} - 1} \right)\\
= 1\\
k,\\
3\sqrt 2 \left( {\sqrt {50} - 2\sqrt {18} + \sqrt {98} } \right)\\
= 3\sqrt 2 .\left( {\sqrt {{5^2}.2} - 2\sqrt {{3^2}.2} + \sqrt {{7^2}.2} } \right)\\
= 3\sqrt 2 .\left( {5\sqrt 2 - 2.3\sqrt 2 + 7\sqrt 2 } \right)\\
= 3\sqrt 2 .\left( {5\sqrt 2 - 6\sqrt 2 + 7\sqrt 2 } \right)\\
= 3\sqrt 2 .6\sqrt 2 \\
= 3.6.2\\
= 36\\
l,\\
\sqrt {27} - 3\sqrt {48} + 2\sqrt {108} - \sqrt {{{\left( {2 - \sqrt 3 } \right)}^2}} \\
= \sqrt {{3^2}.3} - 3\sqrt {{4^2}.3} + 2\sqrt {{6^2}.3} - \left| {2 - \sqrt 3 } \right|\\
= 3\sqrt 3 - 3.4\sqrt 3 + 2.6\sqrt 3 - \left( {2 - \sqrt 3 } \right)\\
= 3\sqrt 3 - 12\sqrt 3 + 12\sqrt 3 - 2 + \sqrt 3 \\
= 4\sqrt 3 - 2\\
m,\\
{\left( {\sqrt {\sqrt 5 + 3} - \sqrt {3 - \sqrt 5 } } \right)^2}\\
= {\sqrt {\sqrt 5 + 3} ^2} - 2.\sqrt {\sqrt 5 + 3} .\sqrt {3 - \sqrt 5 } + {\sqrt {3 - \sqrt 5 } ^2}\\
= \left( {\sqrt 5 + 3} \right) - 2\sqrt {\left( {3 + \sqrt 5 } \right)\left( {3 - \sqrt 5 } \right)} + \left( {3 - \sqrt 5 } \right)\\
= \sqrt 5 + 3 - 2\sqrt {{3^2} - {{\sqrt 5 }^2}} + 3 - \sqrt 5 \\
= 6 - 2\sqrt {9 - 5} \\
= 6 - 2\sqrt 4 \\
= 6 - 2.2\\
= 2
\end{array}\)
