Đề bài
Rút gọn các biểu thức sau
a) \(\left( {\sqrt 8 - 3\sqrt 2 + \sqrt {10} } \right)\sqrt 2 - \sqrt 5 \)
b) \(0,2\sqrt {{{\left( { - 10} \right)}^2}.3} + 2\sqrt {{{\left( {\sqrt 3 - \sqrt 5 } \right)}^2}} \)
c) \(\left( {\dfrac{1}{2}\sqrt {\dfrac{1}{2}} - \dfrac{3}{2}\sqrt 2 + \dfrac{4}{5}\sqrt {200} } \right):\dfrac{1}{8}\)
d) \(2\sqrt {{{\left( {\sqrt 2 - 3} \right)}^2}} + \sqrt {2{{\left( { - 3} \right)}^2}} - 5\sqrt {{{\left( { - 1} \right)}^4}} \)
Phương pháp giải - Xem chi tiết
Sử dụng công thức: \(\begin{array}{l}
\sqrt {AB} = \sqrt A .\sqrt B \,\,\left( {A \ge 0,B \ge 0} \right)\\
\sqrt {{A^2}} = \left| A \right|
\end{array}\)
\(\sqrt {\dfrac{A}{B}} = \dfrac{1}{{\left| B \right|}}\sqrt {AB} ;\)\(\dfrac{A}{{\sqrt B }} = \dfrac{{A\sqrt B }}{B}\left( {B > 0} \right)\)
Lời giải chi tiết
a) \(\left( {\sqrt 8 - 3\sqrt 2 + \sqrt {10} } \right)\sqrt 2 - \sqrt 5 \)\( = \sqrt {8.2} - 3\sqrt {2.2} + \sqrt {20} - \sqrt 5 \)
\( = 4 - 3.2 + 2\sqrt 5 - \sqrt 5 \) \( = \sqrt 5 - 2\)
b) \(0,2\sqrt {{{\left( { - 10} \right)}^2}.3} + 2\sqrt {{{\left( {\sqrt 3 - \sqrt 5 } \right)}^2}} \)
\( = 0,2\left| { - 10} \right|\sqrt 3 + 2\left| {\sqrt 3 - \sqrt 5 } \right|\)
\( = 2\sqrt 3 + 2\left( {\sqrt 5 - \sqrt 3 } \right)\) \( = 2\sqrt 3 + 2\sqrt 5 - 2\sqrt 3 = 2\sqrt 5 \)
c) \(\left( {\dfrac{1}{2}\sqrt {\dfrac{1}{2}} - \dfrac{3}{2}\sqrt 2 + \dfrac{4}{5}\sqrt {200} } \right):\dfrac{1}{8}\)
\( = \left( {\dfrac{1}{2} \cdot \dfrac{1}{2}\sqrt 2 - \dfrac{3}{2}\sqrt 2 + \dfrac{4}{5} \cdot 10\sqrt 2 } \right) \cdot 8\)
\( = 2\sqrt 2 - 12\sqrt 2 + 64\sqrt 2 = 54\sqrt 2 \)
d) \(2\sqrt {{{\left( {\sqrt 2 - 3} \right)}^2}} + \sqrt {2{{\left( { - 3} \right)}^2}} - 5\sqrt {{{\left( { - 1} \right)}^4}} \)
\( = 2\left| {\sqrt 2 - 3} \right| + \left| { - 3} \right|\sqrt 2 - 5\left| {{{\left( { - 1} \right)}^2}} \right|\)
\( = 2\left( {3 - \sqrt 2 } \right) + 3\sqrt 2 - 5\)
\( = 6 - 2\sqrt 2 + 3\sqrt 2 - 5\)
\( = \sqrt 2 + 1\)
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