Câu hỏi/bài tập:
Rút gọn biểu thức
a) \(\left( {5\sqrt {\frac{1}{5}} - \frac{1}{2}\sqrt {20} + \sqrt 5 } \right)\sqrt 5 \)
b) \(\left( {\sqrt {\frac{1}{7}} - \sqrt {\frac{9}{7}} + \sqrt 7 } \right):\sqrt 7 \)
c) \({\left( {\sqrt {\frac{2}{3}} - \sqrt {\frac{3}{2}} } \right)^2}\)
d) \(\frac{{\sqrt {{{312}^2} - {{191}^2}} }}{{\sqrt {503} }}\)
e) \(\sqrt {27.{{\left( {1 - \sqrt 3 } \right)}^4}} :3\sqrt {15} \)
g) \(\frac{{\sqrt[3]{{135}}}}{{\sqrt[3]{5}}} - \sqrt[3]{{54}}.\sqrt[3]{4}\)
a), b) Dùng quy tắc nhân đa thức với đơn thức.
c), d) Khai triển hằng đẳng thức.
e) Biến đổi \(\sqrt {27.{{\left( {1 - \sqrt 3 } \right)}^4}} :3\sqrt {15} = 3.\sqrt {3.} {\left( {1 - \sqrt 3 } \right)^2}.\frac{1}{{3\sqrt {15} }}\)
g) Biến đổi \(\frac{{\sqrt[3]{{135}}}}{{\sqrt[3]{5}}} - \sqrt[3]{{54}}.\sqrt[3]{4} = \frac{{\sqrt[3]{5}.\sqrt[3]{{27}}}}{{\sqrt[3]{5}}} - \sqrt[3]{{2.27}}.\sqrt[3]{4}\).
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a) \(\left( {5\sqrt {\frac{1}{5}} - \frac{1}{2}\sqrt {20} + \sqrt 5 } \right)\sqrt 5 \)
\( = \left( {5\frac{1}{{\sqrt 5 }} - \frac{1}{2}.2.\sqrt 5 + \sqrt 5 } \right)\sqrt 5 = 5 - 5 + 5 = 5.\)
b) \(\left( {\sqrt {\frac{1}{7}} - \sqrt {\frac{9}{7}} + \sqrt 7 } \right):\sqrt 7 \)
\( = \left( {\frac{1}{{\sqrt 7 }} - \frac{3}{{\sqrt 7 }} + \sqrt 7 } \right).\frac{1}{{\sqrt 7 }} = \frac{1}{7} - \frac{3}{7} + 1 = \frac{5}{7}.\)
c) \({\left( {\sqrt {\frac{2}{3}} - \sqrt {\frac{3}{2}} } \right)^2} \)
\(= \frac{2}{3} - 2\sqrt {\frac{2}{3}.\frac{3}{2}} + \frac{3}{2} = \frac{{13}}{6} - 2 = \frac{1}{6}\)
d) \(\frac{{\sqrt {{{312}^2} - {{191}^2}} }}{{\sqrt {503} }} \)
\(= \frac{{\sqrt {\left( {312 - 191} \right)\left( {312 + 191} \right)} }}{{\sqrt {503} }}\)
\( = \frac{{\sqrt {121.503} }}{{\sqrt {503} }} = \sqrt {121} = 11\)
e) \(\sqrt {27.{{\left( {1 - \sqrt 3 } \right)}^4}} :3\sqrt {15} \)
\(= 3.\sqrt {3.} {\left( {1 - \sqrt 3 } \right)^2}.\frac{1}{{3\sqrt {15} }} = \frac{{{{\left( {1 - \sqrt 3 } \right)}^2}}}{{\sqrt 5 }}\)
\( = \frac{{\sqrt 5 \left( {1 - 2\sqrt 3 + 3} \right)}}{5} = \frac{{\sqrt 5 \left( {4 - 2\sqrt 3 } \right)}}{5} = \frac{{4\sqrt 5 - 2\sqrt {15} }}{5}\)
g) \(\frac{{\sqrt[3]{{135}}}}{{\sqrt[3]{5}}} - \sqrt[3]{{54}}.\sqrt[3]{4} \)
\(= \frac{{\sqrt[3]{5}.\sqrt[3]{{27}}}}{{\sqrt[3]{5}}} - \sqrt[3]{{2.27}}.\sqrt[3]{4}\)
\( = 3 - 3\sqrt[3]{2}.\sqrt[3]{4} = 3 - 3\sqrt[3]{8} = 3 - 3.2 = - 3\)