Tổng hợp đáp án Chuyên đề 1 rút gọn biểu thức chứa căn
Bài 1 thực hiện phép tính
\(4\sqrt{5}-3\sqrt{125}+2\sqrt{45}\)\(=4\sqrt{5}-3\sqrt{5.25}+2\sqrt{5.9}\)\(=4\sqrt{5}-3.5\sqrt{5}+2.3\sqrt{5}\)\(=4\sqrt{5}-15\sqrt{5}+6\sqrt{5}\)\(=-5\sqrt{5}\)
\(4\sqrt{5}+3\sqrt{125}-2\sqrt{45}\)\(=4\sqrt{5}+3\sqrt{5.25}-2\sqrt{5.9}\)\(=4\sqrt{5}+3.5\sqrt{5}-2.3\sqrt{5}\)\(=4\sqrt{5}+15\sqrt{5}-6\sqrt{5}\)\(=13\sqrt{5}\)
| \(4\sqrt{5}+\sqrt{125}-2\sqrt{45}\) | \(4\sqrt{5}+\sqrt{125}-2\sqrt{45}\)\(=4\sqrt{5}+\sqrt{5.25}-2\sqrt{5.9}\)\(=4\sqrt{5}+.5\sqrt{5}-2.3\sqrt{5}\)\(=4\sqrt{5}+5\sqrt{5}-6\sqrt{5}\)\(=3\sqrt{5}\) |
| \(2\sqrt{5}+2\sqrt{125}-2\sqrt{45}\) | \(2\sqrt{5}+2\sqrt{125}-2\sqrt{45}\)\(=2\sqrt{5}+2\sqrt{5.25}-2\sqrt{5.9}\)\(=2\sqrt{5}+2.5\sqrt{5}-2.3\sqrt{5}\)\(=2\sqrt{5}+10\sqrt{5}-6\sqrt{5}\)\(=6\sqrt{5}\) |
| \(4\sqrt{3}+2\sqrt{75}-3\sqrt{27}\) | \(4\sqrt{3}+2\sqrt{75}-3\sqrt{27}\)\(=4\sqrt{3}+2\sqrt{3.25}-3\sqrt{3.9}\)\(=4\sqrt{3}+2.5\sqrt{3}-3.3\sqrt{3}\)\(=4\sqrt{3}+10\sqrt{3}-9\sqrt{3}\)\(=5\sqrt{3}\) |
| \(2\sqrt{6}+5\sqrt{384}-2\sqrt{54}\) | \(2\sqrt{6}+5\sqrt{384}-2\sqrt{54}\)\(=2\sqrt{6}+5\sqrt{6.64}-2\sqrt{6.9}\)\(=2\sqrt{6}+5.8\sqrt{6}-2.3\sqrt{6}\)\(=2\sqrt{6}+40\sqrt{6}-6\sqrt{6}\)\(=36\sqrt{6}\) |
| \(4\sqrt{5}+2\sqrt{125}-8\sqrt{125}\) | \(4\sqrt{5}+2\sqrt{125}-8\sqrt{125}\)\(=4\sqrt{5}+2\sqrt{5.25}-8\sqrt{5.25}\)\(=4\sqrt{5}+2.5\sqrt{5}-8.5\sqrt{5}\)\(=4\sqrt{5}+10\sqrt{5}-40\sqrt{5}\)\(=-26\sqrt{5}\) |
| \(2\sqrt{7}+5\sqrt{175}-8\sqrt{63}\) | \(2\sqrt{7}+5\sqrt{175}-8\sqrt{63}\)\(=2\sqrt{7}+5\sqrt{7.25}-8\sqrt{7.9}\)\(=2\sqrt{7}+5.5\sqrt{7}-8.3\sqrt{7}\)\(=2\sqrt{7}+25\sqrt{7}-24\sqrt{7}\)\(=3\sqrt{7}\) |
| \(4\sqrt{2}+4\sqrt{50}-5\sqrt{98}\) | \(4\sqrt{2}+4\sqrt{50}-5\sqrt{98}\)\(=4\sqrt{2}+4\sqrt{2.25}-5\sqrt{2.49}\)\(=4\sqrt{2}+4.5\sqrt{2}-5.7\sqrt{2}\)\(=4\sqrt{2}+20\sqrt{2}-35\sqrt{2}\)\(=-11\sqrt{2}\) |
| \(7\sqrt{1}+2\sqrt{64}-5\sqrt{9}\) | \(7\sqrt{1}+2\sqrt{64}-5\sqrt{9}\)\(=7\sqrt{1}+2\sqrt{1.64}-5\sqrt{1.9}\)\(=7\sqrt{1}+2.8\sqrt{1}-5.3\sqrt{1}\)\(=7\sqrt{1}+16\sqrt{1}-15\sqrt{1}\)\(=8\sqrt{1}\) |
| \(8\sqrt{5}+4\sqrt{320}-5\sqrt{320}\) | \(8\sqrt{5}+4\sqrt{320}-5\sqrt{320}\)\(=8\sqrt{5}+4\sqrt{5.64}-5\sqrt{5.64}\)\(=8\sqrt{5}+4.8\sqrt{5}-5.8\sqrt{5}\)\(=8\sqrt{5}+32\sqrt{5}-40\sqrt{5}\)\(=0\sqrt{5}\) |
| \(2\sqrt{325}+2\sqrt{468}-2\sqrt{117}\) | \(2\sqrt{325}+2\sqrt{468}-2\sqrt{117}\)\(=2\sqrt{13.25}+2\sqrt{13.36}-2\sqrt{13.9}\)\(=2.5\sqrt{13}+2.6\sqrt{13}-2.3\sqrt{13}\)\(=10\sqrt{13}+12\sqrt{13}-6\sqrt{13}\)\(=16\sqrt{13}\) |
| \(\sqrt{5}+3\sqrt{125}-\sqrt{125}\) | \(\sqrt{5}+3\sqrt{125}-\sqrt{125}\)\(=\sqrt{5}+3\sqrt{5.25}-\sqrt{5.25}\)\(=\sqrt{5}+3.5\sqrt{5}-.5\sqrt{5}\)\(=1\sqrt{5}+15\sqrt{5}-5\sqrt{5}\)\(=11\sqrt{5}\) |
| \(2\sqrt{448}+\sqrt{63}-2\sqrt{252}\) | \(2\sqrt{448}+\sqrt{63}-2\sqrt{252}\)\(=2\sqrt{7.64}+\sqrt{7.9}-2\sqrt{7.36}\)\(=2.8\sqrt{7}+.3\sqrt{7}-2.6\sqrt{7}\)\(=16\sqrt{7}+3\sqrt{7}-12\sqrt{7}\)\(=7\sqrt{7}\) |
| \(9\sqrt{125}+2\sqrt{125}-6\sqrt{45}\) | \(9\sqrt{125}+2\sqrt{125}-6\sqrt{45}\)\(=9\sqrt{5.25}+2\sqrt{5.25}-6\sqrt{5.9}\)\(=9.5\sqrt{5}+2.5\sqrt{5}-6.3\sqrt{5}\)\(=45\sqrt{5}+10\sqrt{5}-18\sqrt{5}\)\(=37\sqrt{5}\) |
| \(2\sqrt{5}+3\sqrt{125}-2\sqrt{45}\) | \(2\sqrt{5}+3\sqrt{125}-2\sqrt{45}\)\(=2\sqrt{5}+3\sqrt{5.25}-2\sqrt{5.9}\)\(=2\sqrt{5}+3.5\sqrt{5}-2.3\sqrt{5}\)\(=2\sqrt{5}+15\sqrt{5}-6\sqrt{5}\)\(=11\sqrt{5}\) |
| \(6\sqrt{5}+3\sqrt{125}-6\sqrt{180}\) | \(6\sqrt{5}+3\sqrt{125}-6\sqrt{180}\)\(=6\sqrt{5}+3\sqrt{5.25}-6\sqrt{5.36}\)\(=6\sqrt{5}+3.5\sqrt{5}-6.6\sqrt{5}\)\(=6\sqrt{5}+15\sqrt{5}-36\sqrt{5}\)\(=-15\sqrt{5}\) |
| \(2\sqrt{5}+6\sqrt{125}-2\sqrt{45}\) | \(2\sqrt{5}+6\sqrt{125}-2\sqrt{45}\)\(=2\sqrt{5}+6\sqrt{5.25}-2\sqrt{5.9}\)\(=2\sqrt{5}+6.5\sqrt{5}-2.3\sqrt{5}\)\(=2\sqrt{5}+30\sqrt{5}-6\sqrt{5}\)\(=26\sqrt{5}\) |
Bài 2. Rút gọn các biểu thức sau:
| \(\sqrt { 8 – 2\sqrt 15 } + \sqrt {5 + 2\sqrt 6 } \) | \(\sqrt { 8 – 2\sqrt 15 } + \sqrt {5 + 2\sqrt 6 } \) \(=\sqrt {3 + 5 – 2\sqrt 3 \sqrt 5} + \sqrt {3 + 2 – 2\sqrt 3 \sqrt 2} \) \(=\sqrt {{{\left( {\sqrt 3 – \sqrt 5 } \right)}^2}} + \sqrt {{{\left( {\sqrt 3 + \sqrt 2 } \right)}^2}} \) \(=\left( {-\sqrt 3 + \sqrt 5 } \right)\)+ \(\left( {\sqrt 3 + \sqrt 2 } \right)\) \(=\sqrt 5 – \sqrt 2 \) |
| \(\sqrt { 7 – 2\sqrt 10 } + \sqrt {7 + 2\sqrt 10 } \) | \(\sqrt { 7 – 2\sqrt 10 } + \sqrt {7 + 2\sqrt 10 } \) \(=\sqrt {5 + 2 – 2\sqrt 5 \sqrt 2} + \sqrt {5 + 2 – 2\sqrt 5 \sqrt 2} \) \(=\sqrt {{{\left( {\sqrt 5 – \sqrt 2 } \right)}^2}} + \sqrt {{{\left( {\sqrt 5 + \sqrt 2 } \right)}^2}} \) \(=\left( {\sqrt 5 – \sqrt 2 } \right)\)+ \(\left( {\sqrt 5 + \sqrt 2 } \right)\) \(=2\sqrt 5 – \sqrt 2 + \sqrt 2 \) |
| \(\sqrt { 9 – 2\sqrt 14 } + \sqrt {5 + 2\sqrt 6 } \) | \(\sqrt { 9 – 2\sqrt 14 } + \sqrt {5 + 2\sqrt 6 } \) \(=\sqrt {2 + 7 – 2\sqrt 2 \sqrt 7} + \sqrt {2 + 3 – 2\sqrt 2 \sqrt 3} \) \(=\sqrt {{{\left( {\sqrt 2 – \sqrt 7 } \right)}^2}} + \sqrt {{{\left( {\sqrt 2 + \sqrt 3 } \right)}^2}} \) \(=\left( {-\sqrt 2 + \sqrt 7 } \right)\)+ \(\left( {\sqrt 2 + \sqrt 3 } \right)\) \(=\sqrt 7 – \sqrt 3 \) |
| \(\sqrt { 8 – 2\sqrt 15 } + \sqrt {10 + 2\sqrt 21 } \) | \(\sqrt { 8 – 2\sqrt 15 } + \sqrt {10 + 2\sqrt 21 } \) \(=\sqrt {3 + 5 – 2\sqrt 3 \sqrt 5} + \sqrt {3 + 7 – 2\sqrt 3 \sqrt 7} \) \(=\sqrt {{{\left( {\sqrt 3 – \sqrt 5 } \right)}^2}} + \sqrt {{{\left( {\sqrt 3 + \sqrt 7 } \right)}^2}} \) \(=\left( {-\sqrt 3 + \sqrt 5 } \right)\)+ \(\left( {\sqrt 3 + \sqrt 7 } \right)\) \(=\sqrt 5 – \sqrt 7 \) |
| \(\sqrt { 7 – 2\sqrt 10 } + \sqrt {9 + 2\sqrt 14 } \) | \(\sqrt { 7 – 2\sqrt 10 } + \sqrt {9 + 2\sqrt 14 } \) \(=\sqrt {2 + 5 – 2\sqrt 2 \sqrt 5} + \sqrt {2 + 7 – 2\sqrt 2 \sqrt 7} \) \(=\sqrt {{{\left( {\sqrt 2 – \sqrt 5 } \right)}^2}} + \sqrt {{{\left( {\sqrt 2 + \sqrt 7 } \right)}^2}} \) \(=\left( {-\sqrt 2 + \sqrt 5 } \right)\)+ \(\left( {\sqrt 2 + \sqrt 7 } \right)\) \(=\sqrt 5 – \sqrt 7 \) |
| \(\sqrt { 12 – 2\sqrt 35 } + \sqrt {9 + 2\sqrt 14 } \) | \(\sqrt { 12 – 2\sqrt 35 } + \sqrt {9 + 2\sqrt 14 } \) \(=\sqrt {7 + 5 – 2\sqrt 7 \sqrt 5} + \sqrt {7 + 2 – 2\sqrt 7 \sqrt 2} \) \(=\sqrt {{{\left( {\sqrt 7 – \sqrt 5 } \right)}^2}} + \sqrt {{{\left( {\sqrt 7 + \sqrt 2 } \right)}^2}} \) \(=\left( {\sqrt 7 – \sqrt 5 } \right)\)+ \(\left( {\sqrt 7 + \sqrt 2 } \right)\) \(=2\sqrt 7 – \sqrt 5 + \sqrt 2 \) |
| \(\sqrt { 5 – 2\sqrt 6 } + \sqrt {5 + 2\sqrt 6 } \) | \(\sqrt { 5 – 2\sqrt 6 } + \sqrt {5 + 2\sqrt 6 } \) \(=\sqrt {3 + 2 – 2\sqrt 3 \sqrt 2} + \sqrt {3 + 2 – 2\sqrt 3 \sqrt 2} \) \(=\sqrt {{{\left( {\sqrt 3 – \sqrt 2 } \right)}^2}} + \sqrt {{{\left( {\sqrt 3 + \sqrt 2 } \right)}^2}} \) \(=\left( {\sqrt 3 – \sqrt 2 } \right)\)+ \(\left( {\sqrt 3 + \sqrt 2 } \right)\) \(=2\sqrt 3 – \sqrt 2 + \sqrt 2 \) |
| \(\sqrt { 7 – 2\sqrt 10 } + \sqrt {7 + 2\sqrt 10 } \) | \(\sqrt { 7 – 2\sqrt 10 } + \sqrt {7 + 2\sqrt 10 } \) \(=\sqrt {5 + 2 – 2\sqrt 5 \sqrt 2} + \sqrt {5 + 2 – 2\sqrt 5 \sqrt 2} \) \(=\sqrt {{{\left( {\sqrt 5 – \sqrt 2 } \right)}^2}} + \sqrt {{{\left( {\sqrt 5 + \sqrt 2 } \right)}^2}} \) \(=\left( {\sqrt 5 – \sqrt 2 } \right)\)+ \(\left( {\sqrt 5 + \sqrt 2 } \right)\) \(=2\sqrt 5 – \sqrt 2 + \sqrt 2 \) |
| \(\sqrt { 5 – 2\sqrt 6 } + \sqrt {4 + 2\sqrt 4 } \) | \(\sqrt { 5 – 2\sqrt 6 } + \sqrt {4 + 2\sqrt 4 } \) \(=\sqrt {2 + 3 – 2\sqrt 2 \sqrt 3} + \sqrt {2 + 2 – 2\sqrt 2 \sqrt 2} \) \(=\sqrt {{{\left( {\sqrt 2 – \sqrt 3 } \right)}^2}} + \sqrt {{{\left( {\sqrt 2 + \sqrt 2 } \right)}^2}} \) \(=\left( {-\sqrt 2 + \sqrt 3 } \right)\)+ \(\left( {\sqrt 2 + \sqrt 2 } \right)\) \(=\sqrt 3 – \sqrt 2 \) |
| \(\sqrt { 10 – 2\sqrt 21 } + \sqrt {5 + 2\sqrt 6 } \) | \(\sqrt { 10 – 2\sqrt 21 } + \sqrt {5 + 2\sqrt 6 } \) \(=\sqrt {3 + 7 – 2\sqrt 3 \sqrt 7} + \sqrt {3 + 2 – 2\sqrt 3 \sqrt 2} \) \(=\sqrt {{{\left( {\sqrt 3 – \sqrt 7 } \right)}^2}} + \sqrt {{{\left( {\sqrt 3 + \sqrt 2 } \right)}^2}} \) \(=\left( {-\sqrt 3 + \sqrt 7 } \right)\)+ \(\left( {\sqrt 3 + \sqrt 2 } \right)\) \(=\sqrt 7 – \sqrt 2 \) |
| \(\sqrt { 5 – 2\sqrt 6 } + \sqrt {9 + 2\sqrt 14 } \) | \(\sqrt { 5 – 2\sqrt 6 } + \sqrt {9 + 2\sqrt 14 } \) \(=\sqrt {2 + 3 – 2\sqrt 2 \sqrt 3} + \sqrt {2 + 7 – 2\sqrt 2 \sqrt 7} \) \(=\sqrt {{{\left( {\sqrt 2 – \sqrt 3 } \right)}^2}} + \sqrt {{{\left( {\sqrt 2 + \sqrt 7 } \right)}^2}} \) \(=\left( {-\sqrt 2 + \sqrt 3 } \right)\)+ \(\left( {\sqrt 2 + \sqrt 7 } \right)\) \(=\sqrt 3 – \sqrt 7 \) |
| \(\sqrt { 9 – 2\sqrt 14 } + \sqrt {10 + 2\sqrt 21 } \) | \(\sqrt { 9 – 2\sqrt 14 } + \sqrt {10 + 2\sqrt 21 } \) \(=\sqrt {7 + 2 – 2\sqrt 7 \sqrt 2} + \sqrt {7 + 3 – 2\sqrt 7 \sqrt 3} \) \(=\sqrt {{{\left( {\sqrt 7 – \sqrt 2 } \right)}^2}} + \sqrt {{{\left( {\sqrt 7 + \sqrt 3 } \right)}^2}} \) \(=\left( {\sqrt 7 – \sqrt 2 } \right)\)+ \(\left( {\sqrt 7 + \sqrt 3 } \right)\) \(=2\sqrt 7 – \sqrt 2 + \sqrt 3 \) |
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