Tính giá trị biểu thức A= (√(45+27√2) +√(45-27√2))/(√(5+3√2)-√(5-3√2))+(√(3+√2)+√(3-√2))/(√(3+√2)-√(3-√2))

Tính giá trị biểu thức A= (√(45+27√2) +√(45-27√2))/(√(5+3√2)-√(5-3√2))+(√(3+√2)+√(3-√2))/(√(3+√2)-√(3-√2))

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  1. \[\begin{array}{l}
    A = \frac{{\sqrt {45 + 27\sqrt 2 } + \sqrt {45 – 27\sqrt 2 } }}{{\sqrt {5 + 3\sqrt 2 } – \sqrt {5 – 3\sqrt 2 } }} + \frac{{\sqrt {3 + \sqrt 2 } + \sqrt {3 – \sqrt 2 } }}{{\sqrt {3 + \sqrt 2 } – \sqrt {3 – \sqrt 2 } }}\\
    = \frac{{\sqrt {9\left( {5 + 3\sqrt 2 } \right)} + \sqrt {9\left( {5 – 3\sqrt 2 } \right)} }}{{\sqrt {5 + 3\sqrt 2 } – \sqrt {5 – 3\sqrt 2 } }} + \frac{{\sqrt {3 + \sqrt 2 } + \sqrt {3 – \sqrt 2 } }}{{\sqrt {3 + \sqrt 2 } – \sqrt {3 – \sqrt 2 } }}\\
    = \frac{{3\left( {\sqrt {5 + 3\sqrt 2 } + \sqrt {5 – 3\sqrt 2 } } \right)}}{{\sqrt {5 + 3\sqrt 2 } – \sqrt {5 – 3\sqrt 2 } }} + \frac{{{{\left( {\sqrt {3 + \sqrt 2 } + \sqrt {3 – \sqrt 2 } } \right)}^2}}}{{3 + \sqrt 2 – 3 + \sqrt 2 }}\\
    = \frac{{3{{\left( {\sqrt {5 + 3\sqrt 2 } + \sqrt {5 – 3\sqrt 2 } } \right)}^2}}}{{5 + 3\sqrt 2 – 5 + 3\sqrt 2 }} + \frac{{3 + \sqrt 2 + 3 – \sqrt 2 + 2\sqrt {\left( {3 + \sqrt 2 } \right)\left( {3 – \sqrt 2 } \right)} }}{{2\sqrt 2 }}\\
    = \frac{{3\left( {5 + 3\sqrt 2 + 5 – 3\sqrt 2 + 2\sqrt {\left( {5 + 3\sqrt 2 } \right)\left( {5 – 3\sqrt 2 } \right)} } \right)}}{{6\sqrt 2 }} + \frac{{6 + 2\sqrt {9 – 2} }}{{2\sqrt 2 }}\\
    = \frac{{3\left( {10 + 2\sqrt {25 – 18} } \right)}}{{6\sqrt 2 }} + \frac{{3 + \sqrt 7 }}{{\sqrt 2 }}\\
    = \frac{{5 + \sqrt 7 }}{{\sqrt 2 }} + \frac{{3 + \sqrt 7 }}{{\sqrt 2 }} = \frac{{5 + \sqrt 7 + 3 + \sqrt 7 }}{{\sqrt 2 }}\\
    = \frac{{8 + 2\sqrt 7 }}{{\sqrt 2 }} = 2\sqrt 2 + \sqrt {14} .
    \end{array}\]

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