Question

Đã trả lời bởi chuyên gia

Answer 1

$\sqrt{11 - 6 \sqrt{2}} + 3 \sqrt{2} = \sqrt{9 - 2.3 . \sqrt{2} + 2} + 3 \sqrt{2} = \sqrt{{(3 - \sqrt{2})}^{2}} + 3 \sqrt{2} = |3 - \sqrt{2}| + 3 \sqrt{2} = 3 - \sqrt{2} + 3 \sqrt{2} (d o \sqrt{9} = 3 > \sqrt{2}) = 3 + 2 \sqrt{2}$

$\sqrt{6 - 2 \sqrt{5}} + \sqrt{6 + 2 \sqrt{5}} = \sqrt{5 - 2 \sqrt{5} . 1 + 1} + \sqrt{5 + 2 \sqrt{5} . 1 + 1} = \sqrt{{(\sqrt{5} - 1)}^{2}} + \sqrt{{(\sqrt{5} + 1)}^{2}} = |\sqrt{5} - 1| + \sqrt{5} + 1 = \sqrt{5} - 1 + \sqrt{5} + 1 (d o \sqrt{5} > 1) = 2 \sqrt{5}$

$\sqrt{7 + 4 \sqrt{3}} - \sqrt{7 - 4 \sqrt{3}} = \sqrt{4 + 2 . \sqrt{3} . 2 + \sqrt{3}} - \sqrt{4 - 2.2 . \sqrt{3} + 3} = \sqrt{{(2 + \sqrt{3})}^{2}} - \sqrt{{(2 - \sqrt{3})}^{2}} = 2 + \sqrt{3} - |2 - \sqrt{3}| = 2 + \sqrt{3} - 2 + \sqrt{3} (d o 2 > \sqrt{3}) = 2 \sqrt{3}$

$\sqrt{6 + 2 \sqrt{5 - \sqrt{13 + 4 \sqrt{3}}}} = \sqrt{6 + 2 \sqrt{5 - \sqrt{12 + 2.2 \sqrt{3} + 1}}} = \sqrt{6 + 2 \sqrt{5 - \sqrt{{(2 \sqrt{3} + 1)}^{2}}}} = \sqrt{6 + 2 \sqrt{5 - |2 \sqrt{3} + 1|}} = \sqrt{6 + 2 \sqrt{5 - 2 \sqrt{3} - 1}} (d o 2 \sqrt{3} > 1) = \sqrt{6 + 2 \sqrt{4 - 2 \sqrt{3}}} = \sqrt{6 + 2 \sqrt{3 - 2 \sqrt{3} + 1}} = \sqrt{6 + 2 \sqrt{{(\sqrt{3} - 1)}^{2}}} = \sqrt{6 + 2 (\sqrt{3} - 1)} (d o \sqrt{3} > 1) = \sqrt{4 + 2 \sqrt{3}} = \sqrt{3 + 2 \sqrt{3} + 1} = \sqrt{{(\sqrt{3} + 1)}^{2}} = \sqrt{3} + 1$ .

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Answer 2

$\sqrt{11 - 6 \sqrt{2}} + 3 \sqrt{2} = \sqrt{9 - 2.3 . \sqrt{2} + 2} + 3 \sqrt{2} = \sqrt{{(3 - \sqrt{2})}^{2}} + 3 \sqrt{2} = |3 - \sqrt{2}| + 3 \sqrt{2} = 3 - \sqrt{2} + 3 \sqrt{2} (d o \sqrt{9} = 3 > \sqrt{2}) = 3 + 2 \sqrt{2}$

$\sqrt{6 - 2 \sqrt{5}} + \sqrt{6 + 2 \sqrt{5}} = \sqrt{5 - 2 \sqrt{5} . 1 + 1} + \sqrt{5 + 2 \sqrt{5} . 1 + 1} = \sqrt{{(\sqrt{5} - 1)}^{2}} + \sqrt{{(\sqrt{5} + 1)}^{2}} = |\sqrt{5} - 1| + \sqrt{5} + 1 = \sqrt{5} - 1 + \sqrt{5} + 1 (d o \sqrt{5} > 1) = 2 \sqrt{5}$

$\sqrt{7 + 4 \sqrt{3}} - \sqrt{7 - 4 \sqrt{3}} = \sqrt{4 + 2 . \sqrt{3} . 2 + \sqrt{3}} - \sqrt{4 - 2.2 . \sqrt{3} + 3} = \sqrt{{(2 + \sqrt{3})}^{2}} - \sqrt{{(2 - \sqrt{3})}^{2}} = 2 + \sqrt{3} - |2 - \sqrt{3}| = 2 + \sqrt{3} - 2 + \sqrt{3} (d o 2 > \sqrt{3}) = 2 \sqrt{3}$

$\sqrt{6 + 2 \sqrt{5 - \sqrt{13 + 4 \sqrt{3}}}} = \sqrt{6 + 2 \sqrt{5 - \sqrt{12 + 2.2 \sqrt{3} + 1}}} = \sqrt{6 + 2 \sqrt{5 - \sqrt{{(2 \sqrt{3} + 1)}^{2}}}} = \sqrt{6 + 2 \sqrt{5 - |2 \sqrt{3} + 1|}} = \sqrt{6 + 2 \sqrt{5 - 2 \sqrt{3} - 1}} (d o 2 \sqrt{3} > 1) = \sqrt{6 + 2 \sqrt{4 - 2 \sqrt{3}}} = \sqrt{6 + 2 \sqrt{3 - 2 \sqrt{3} + 1}} = \sqrt{6 + 2 \sqrt{{(\sqrt{3} - 1)}^{2}}} = \sqrt{6 + 2 (\sqrt{3} - 1)} (d o \sqrt{3} > 1) = \sqrt{4 + 2 \sqrt{3}} = \sqrt{3 + 2 \sqrt{3} + 1} = \sqrt{{(\sqrt{3} + 1)}^{2}} = \sqrt{3} + 1$ .

Answer 3

\(\begin{array}{l}
\sqrt {11 - 6\sqrt 2 } + 3\sqrt 2 \\
= \sqrt {9 - 2.3.\sqrt 2 + 2} + 3\sqrt 2 \\
= \sqrt {{{(3 - \sqrt 2 )}^2}} + 3\sqrt 2 \\
= |3 - \sqrt 2 | + 3\sqrt 2 \\
= 3 - \sqrt 2 + 3\sqrt 2 (do:3 - \sqrt 2 > 0)\\
= 3 + 2\sqrt 2 \\
\sqrt {6 - 2\sqrt 5 } + \sqrt {6 + 2\sqrt 5 } \\
= \sqrt {5 - 2\sqrt 5 .1 + 1} + \sqrt {5 + 2\sqrt 5 .1 + 1} \\
= \sqrt {{{(\sqrt 5 - 1)}^2}} + \sqrt {{{(\sqrt 5 + 1)}^2}} \\
= |\sqrt 5 - 1| + |\sqrt 5 + 1|\\
= \sqrt 5 - 1 + \sqrt 5 + 1(do:\sqrt 5 - 1 > 0;\sqrt 5 + 1 > 0)\\
= 2\sqrt 5 \\
\sqrt {7 + 4\sqrt 3 } - \sqrt {7 - 4\sqrt 3 } \\
= \sqrt {4 + 2.2\sqrt 3 + 3} - \sqrt {4 - 2.2\sqrt 3 + 3} \\
= \sqrt {{{(2 + \sqrt 3 )}^2}} - \sqrt {{{(2 - \sqrt 3 )}^2}} \\
= |2 + \sqrt 3 | - |2 - \sqrt 3 |\\
= 2 + \sqrt 3 - (2 - \sqrt 3 )(do:2 + \sqrt 3 > 0;2 - \sqrt 3 > 0)\\
= 2 + \sqrt 3 - 2 + \sqrt 3 \\
= 2\sqrt 3 \\
\sqrt {6 + 2\sqrt {5 - \sqrt {13 + 4\sqrt 3 } } } \\
= \sqrt {6 + 2\sqrt {5 - \sqrt {12 + 2.2\sqrt 3 .1 + 1} } } \\
= \sqrt {6 + 2\sqrt {5 - \sqrt {{{(2\sqrt 3 + 1)}^2}} } } \\
= \sqrt {6 + 2\sqrt {5 - |2\sqrt 3 + 1|} } \\
= \sqrt {6 + 2\sqrt {5 - (2\sqrt 3 + 1)} } (do:2\sqrt 3 + 1 > 0)\\
= \sqrt {6 + 2\sqrt {5 - 2\sqrt 3 - 1} } \\
= \sqrt {6 + 2\sqrt {4 - 2\sqrt 3 } } \\
= \sqrt {6 + 2\sqrt {3 - 2\sqrt 3 + 1} } \\
= \sqrt {6 + 2\sqrt {{{(\sqrt 3 - 1)}^2}} } \\
= \sqrt {6 + 2|\sqrt 3 - 1|} \\
= \sqrt {6 + 2(\sqrt 3 - 1)} (do:\sqrt 3 - 1 > 0)\\
= \sqrt {6 + 2\sqrt 3 - 2} \\
= \sqrt {3 + 2\sqrt 3 + 1} \\
= \sqrt {{{(\sqrt 3 + 1)}^2}} \\
= |\sqrt 3 + 1|\\
= \sqrt 3 + 1(do:\sqrt 3 + 1 > 0)
\end{array}\)

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Answer 4

\(\begin{array}{l}
\sqrt {11 - 6\sqrt 2 } + 3\sqrt 2 \\
= \sqrt {9 - 2.3.\sqrt 2 + 2} + 3\sqrt 2 \\
= \sqrt {{{(3 - \sqrt 2 )}^2}} + 3\sqrt 2 \\
= |3 - \sqrt 2 | + 3\sqrt 2 \\
= 3 - \sqrt 2 + 3\sqrt 2 (do:3 - \sqrt 2 > 0)\\
= 3 + 2\sqrt 2 \\
\sqrt {6 - 2\sqrt 5 } + \sqrt {6 + 2\sqrt 5 } \\
= \sqrt {5 - 2\sqrt 5 .1 + 1} + \sqrt {5 + 2\sqrt 5 .1 + 1} \\
= \sqrt {{{(\sqrt 5 - 1)}^2}} + \sqrt {{{(\sqrt 5 + 1)}^2}} \\
= |\sqrt 5 - 1| + |\sqrt 5 + 1|\\
= \sqrt 5 - 1 + \sqrt 5 + 1(do:\sqrt 5 - 1 > 0;\sqrt 5 + 1 > 0)\\
= 2\sqrt 5 \\
\sqrt {7 + 4\sqrt 3 } - \sqrt {7 - 4\sqrt 3 } \\
= \sqrt {4 + 2.2\sqrt 3 + 3} - \sqrt {4 - 2.2\sqrt 3 + 3} \\
= \sqrt {{{(2 + \sqrt 3 )}^2}} - \sqrt {{{(2 - \sqrt 3 )}^2}} \\
= |2 + \sqrt 3 | - |2 - \sqrt 3 |\\
= 2 + \sqrt 3 - (2 - \sqrt 3 )(do:2 + \sqrt 3 > 0;2 - \sqrt 3 > 0)\\
= 2 + \sqrt 3 - 2 + \sqrt 3 \\
= 2\sqrt 3 \\
\sqrt {6 + 2\sqrt {5 - \sqrt {13 + 4\sqrt 3 } } } \\
= \sqrt {6 + 2\sqrt {5 - \sqrt {12 + 2.2\sqrt 3 .1 + 1} } } \\
= \sqrt {6 + 2\sqrt {5 - \sqrt {{{(2\sqrt 3 + 1)}^2}} } } \\
= \sqrt {6 + 2\sqrt {5 - |2\sqrt 3 + 1|} } \\
= \sqrt {6 + 2\sqrt {5 - (2\sqrt 3 + 1)} } (do:2\sqrt 3 + 1 > 0)\\
= \sqrt {6 + 2\sqrt {5 - 2\sqrt 3 - 1} } \\
= \sqrt {6 + 2\sqrt {4 - 2\sqrt 3 } } \\
= \sqrt {6 + 2\sqrt {3 - 2\sqrt 3 + 1} } \\
= \sqrt {6 + 2\sqrt {{{(\sqrt 3 - 1)}^2}} } \\
= \sqrt {6 + 2|\sqrt 3 - 1|} \\
= \sqrt {6 + 2(\sqrt 3 - 1)} (do:\sqrt 3 - 1 > 0)\\
= \sqrt {6 + 2\sqrt 3 - 2} \\
= \sqrt {3 + 2\sqrt 3 + 1} \\
= \sqrt {{{(\sqrt 3 + 1)}^2}} \\
= |\sqrt 3 + 1|\\
= \sqrt 3 + 1(do:\sqrt 3 + 1 > 0)
\end{array}\)

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