Câu 59: Rút gọn các biểu thức:
a) \(\left( {2\sqrt 3 + \sqrt 5 } \right)\sqrt 3 – \sqrt {60} \);
b) \(\left( {5\sqrt 2 + 2\sqrt 5 } \right)\sqrt 5 – \sqrt {250} \);
c) \(\left( {\sqrt {28} – \sqrt {12} – \sqrt 7 } \right)\sqrt 7 + 2\sqrt {21} \);
d) \(\left( {\sqrt {99} – \sqrt {18} – \sqrt {11} } \right)\sqrt {11} + 3\sqrt {22} \).
\(\eqalign{
& a)\,\left( {2\sqrt 3 + \sqrt 5 } \right)\sqrt 3 – \sqrt {60} \cr
& = 2\sqrt {{3^2}} + \sqrt {15} – \sqrt {4.15} \cr} \)
\( = 6 + \sqrt {15} – 2\sqrt {15} = 6 – \sqrt {15} \)
\(\eqalign{
& b)\,\left( {5\sqrt 2 + 2\sqrt 5 } \right)\sqrt 5 – \sqrt {250} \cr
& = 5\sqrt {10} + 2\sqrt {{5^2}} – \sqrt {25.10} \cr} \)
\( = 5\sqrt {10} + 10 – 5\sqrt {10} = 10\)
\(\eqalign{
& c)\,\left( {\sqrt {28} – \sqrt {12} – \sqrt 7 } \right)\sqrt 7 + 2\sqrt {21} \cr
& = \left( {\sqrt {4.7} – \sqrt {4.3} – \sqrt 7 } \right)\sqrt 7 + 2\sqrt {21} \cr} \)
\( = \left( {2\sqrt 7 – 2\sqrt 3 – \sqrt 7 } \right)\sqrt 7 + 2\sqrt {21} \)
\( = 2\sqrt {{7^2}} – 2\sqrt {21} – \sqrt {{7^2}} + 2\sqrt {21} \)
\( = 14 – 7 = 7\)
\(\eqalign{
& d)\,\left( {\sqrt {99} – \sqrt {18} – \sqrt {11} } \right)\sqrt {11} + 3\sqrt {22} \cr
& = \left( {\sqrt {9.11} – \sqrt {9.2} – \sqrt {11} } \right)\sqrt {11} + 3\sqrt {22} \cr} \)
\( = \left( {3\sqrt {11} – 3\sqrt 2 – \sqrt {11} } \right)\sqrt {11} + 3\sqrt {22} \)
\( = 3\sqrt {{{11}^2}} – 3\sqrt {22} – \sqrt {{{11}^2}} + 3\sqrt {22} \)
\( = 33 – 11 = 22\)
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Câu 60: Rút gọn các biểu thức:
a) \(2\sqrt {40\sqrt {12} } – 2\sqrt {\sqrt {75} } – 3\sqrt {5\sqrt {48} } \);
b) \(2\sqrt {8\sqrt 3 } – 2\sqrt {5\sqrt 3 } – 3\sqrt {20\sqrt 3 } \).
a) \(2\sqrt {40\sqrt {12} } – 2\sqrt {\sqrt {75} } – 3\sqrt {5\sqrt {48} } \)
\( = 2\sqrt {40\sqrt {4.3} } – 2\sqrt {\sqrt {25.3} } – 3\sqrt {5\sqrt {16.3} } \)
\( = 2\sqrt {80\sqrt 3 } – 2\sqrt {5\sqrt 3 } – 3\sqrt {5.4\sqrt 3 } \)
\( = 2\sqrt {16.5\sqrt 3 } – 2\sqrt {5\sqrt 3 } – 3\sqrt {5.4\sqrt 3 } \)
\( = 8\sqrt {5\sqrt 3 } – 2\sqrt {5\sqrt 3 } – 6\sqrt {5\sqrt 3 } = 0\)
\(\eqalign{
& b)\,2\sqrt {8\sqrt 3 } – 2\sqrt {5\sqrt 3 } – 3\sqrt {20\sqrt 3 } \cr
& = 2\sqrt {4.2\sqrt 3 } – 2\sqrt {5\sqrt 3 } – 3\sqrt {4.5\sqrt 3 } \cr} \)
\(\eqalign{
& = 4\sqrt {2\sqrt 3 } – 2\sqrt {5\sqrt 3 } – 6\sqrt {5\sqrt 3 } \cr
& = 4\sqrt 2 – 8\sqrt {5\sqrt 3 } \cr} \)
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Câu 61: Khai triển và rút gọn các biểu thức ( với x và y không âm):
a) \(\left( {1 – \sqrt x } \right)\left( {1 + \sqrt x + x} \right)\);
b) \(\left( {\sqrt x + 2} \right)\left( {x – 2\sqrt x + 4} \right)\);
c) \(\left( {\sqrt x – \sqrt y } \right)\left( {x + y – \sqrt {xy} } \right)\);
d) \(\left( {\sqrt x + \sqrt y } \right)\left( {{x^2} + y – x\sqrt y } \right)\).
\(\eqalign{
& a)\,\left( {1 – \sqrt x } \right)\left( {1 + \sqrt x + x} \right) \cr
& = \left( {1 – \sqrt x } \right)\left[ {1 + 1\sqrt x + {{\left( {\sqrt x } \right)}^2}} \right] \cr} \)
\( = 1 – {\left( {\sqrt x } \right)^3} = 1 – x\sqrt x \) (với \(x \ge 0\))
\(\eqalign{
& b)\,\left( {\sqrt x + 2} \right)\left( {x – 2\sqrt x + 4} \right) \cr
& = \left( {\sqrt x + 2} \right)\left[ {{{\left( {\sqrt x } \right)}^2} – \sqrt x .2 + {2^2}} \right] \cr} \)
\( = {\left( {\sqrt x } \right)^3} + {2^3} = x\sqrt x + 8\) (với \(x \ge 0\))
c) \(\left( {\sqrt x – \sqrt y } \right)\left( {x + y – \sqrt {xy} } \right)\)
\( = \left( {\sqrt x – \sqrt y } \right)\left[ {{{\left( {\sqrt x } \right)}^2} – \sqrt x .\sqrt y + {{\left( {\sqrt y } \right)}^2}} \right]\)
\( = {\left( {\sqrt x } \right)^3} – {\left( {\sqrt y } \right)^3} = x\sqrt x – y\sqrt y \) (với \(x \ge 0\), \(y \ge 0\))
\(\eqalign{
& d)\,\,\left( {\sqrt x + \sqrt y } \right)\left( {{x^2} + y – x\sqrt y } \right) \cr
& = \left( {\sqrt x + \sqrt y } \right)\left[ {{x^2} – x\sqrt y + {{\left( {\sqrt y } \right)}^2}} \right] \cr} \)
\( = {x^3} + {\left( {\sqrt y } \right)^3} = {x^3} + y\sqrt y \) (với \(y \ge 0\))
Câu 62: Khai triển và rút gọn các biểu thức (với x, y không âm):
a) \(\left( {4\sqrt x – \sqrt {2x} } \right)\left( {\sqrt x – \sqrt {2x} } \right)\);
b) \(\left( {2\sqrt x + \sqrt y } \right)\left( {3\sqrt x – 2\sqrt y } \right)\).
a) \(\left( {4\sqrt x – 2\sqrt x } \right)\left( {\sqrt x – \sqrt {2x} } \right)\)
\( = 4\sqrt {{x^2}} – 4\sqrt {2{x^2}} – \sqrt {2{x^2}} + \sqrt {4{x^2}} \)
\(\eqalign{
& = 4x – 4x\sqrt 2 – x\sqrt 2 + 2x \cr
& = 6x – 5x\sqrt 2 \cr} \) (với \(x \ge 0\))
b) \(\left( {2\sqrt x + \sqrt y } \right)\left( {3\sqrt x – 2\sqrt y } \right)\)
\( = 6\sqrt {{x^2}} – 4\sqrt {xy} + 3\sqrt {xy} – 2\sqrt {{y^2}} \)
\( = 6x – \sqrt {xy} – 2y\) (với \(x \ge 0\), \(y \ge 0\))
