Bài tập Rút gọn biểu thức
  • Bài tập tự luyện dạng 6

Câu 1:

Rút gọn các biểu thức sau:

a) \[\sqrt{112}-3\sqrt{175}+\sqrt{252}\] b) \[\sqrt{99}-\frac{1}{4}\sqrt{176}+\frac{1}{5}\sqrt{275}\]

c) \[6\left( \sqrt{\frac{2}{3}}+\sqrt{\frac{3}{2}} \right)\] d) \[3\sqrt{\frac{5}{9}}+5\sqrt{\frac{1}{125}}-9\sqrt{\frac{5}{81}}\]

Câu 2:

Không sử dụng máy tính hãy chứng minh

a) \[A=\frac{2\sqrt{3+2\sqrt{3}-\sqrt{2}+\sqrt{3+2\sqrt{2}}}}{\sqrt{3}-1}-2\sqrt{3}\] là số nguyên.

b) \[B=\frac{\left( 5+2\sqrt{6} \right)\left( 49-20\sqrt{6} \right)\sqrt{5-2\sqrt{6}}}{9\sqrt{3}-11\sqrt{2}}\] là số nguyên.

Câu 3:

Tính:

a) \[\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+…+\frac{1}{\sqrt{99}+\sqrt{100}}.\]

b) \[\frac{1}{\sqrt{1}-\sqrt{2}}-\frac{1}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{4}}-…-\frac{1}{\sqrt{50}-\sqrt{51}}.\]

LỜI GIẢI BÀI TẬP TỰ LUYỆN

Câu 1:

a) \[\sqrt{112}-3\sqrt{175}+\sqrt{252}=4\sqrt{7}-15\sqrt{7}+6\sqrt{7}=-5\sqrt{7}.\]

b) \[\sqrt{99}-\frac{1}{4}\sqrt{176}+\frac{1}{5}\sqrt{275}=3\sqrt{11}-\sqrt{11}+\sqrt{11}=3\sqrt{11}.\]

c) \[6\left( \sqrt{\frac{2}{3}}+\sqrt{\frac{3}{2}} \right)=6.\left( \frac{\sqrt{6}}{3}+\frac{\sqrt{6}}{2} \right)=5\sqrt{6}.\]

d) \[3\sqrt{\frac{5}{9}}+5\sqrt{\frac{1}{125}}-9\sqrt{\frac{5}{81}}=\sqrt{5}+\frac{\sqrt{5}}{5}-\sqrt{5}=\frac{\sqrt{5}}{5}.\]

Câu 2:

  1. Ta có \[A=\frac{2\sqrt{3+2\sqrt{3}-\sqrt{2}+\sqrt{3+2\sqrt{2}}}}{\sqrt{3}-1}-2\sqrt{3}\]

\[=\frac{2\sqrt{3+2\sqrt{3}+1-1-\sqrt{2}+\sqrt{{{\left( \sqrt{2}+1 \right)}^{2}}}}}{\sqrt{3}-1}-2\sqrt{3}\]

\[=\frac{2\sqrt{{{\left( \sqrt{3}+1 \right)}^{2}}-\left( \sqrt{2}+1 \right)+\sqrt{2}+1}}{\sqrt{3}-1}-2\sqrt{3}\]

\[=\frac{2\left( \sqrt{3}+1 \right)}{\sqrt{3}-1}-2\sqrt{3}\]

\[=\frac{2\sqrt{3}+2-2\sqrt{3}\left( \sqrt{3}-1 \right)}{\sqrt{3}-1}\]

\[=\frac{2\sqrt{3}+2-6+2\sqrt{3}}{\sqrt{3}-1}\]

\[=\frac{4\sqrt{3}-4}{\sqrt{3}-1}\]

\[=\frac{4\left( \sqrt{3}-1 \right)}{\sqrt{3}-1}=4\].

Điều phải chứng minh.

  1. Ta có \[B=\frac{\left( 5+2\sqrt{6} \right)\left( 49-20\sqrt{6} \right)\sqrt{5-2\sqrt{6}}}{9\sqrt{3}-11\sqrt{2}}\]

\[=\frac{{{\left( \sqrt{3}+\sqrt{2} \right)}^{2}}\left( 49-20\sqrt{6} \right)\sqrt{{{\left( \sqrt{3}-\sqrt{2} \right)}^{2}}}}{9\sqrt{3}-11\sqrt{2}}\]

\[=\frac{{{\left( \sqrt{3}+\sqrt{2} \right)}^{2}}\left( \sqrt{3}-\sqrt{2} \right)\left( 49-20\sqrt{6} \right)}{9\sqrt{3}-11\sqrt{2}}\]

\[=\frac{\left( \sqrt{3}+\sqrt{2} \right)\left( 3-2 \right)\left( 49-20\sqrt{6} \right)}{9\sqrt{3}-11\sqrt{2}}\]

\[=\frac{49\sqrt{3}+49\sqrt{2}-20\sqrt{18}-20\sqrt{12}}{9\sqrt{3}-11\sqrt{2}}\]

\[=\frac{49\sqrt{3}+49\sqrt{2}-60\sqrt{2}-40\sqrt{3}}{9\sqrt{3}-11\sqrt{2}}\]

\[=\frac{9\sqrt{3}-11\sqrt{2}}{9\sqrt{3}-11\sqrt{2}}=1\] (Điều phải chứng minh).

Câu 3:

a) \[\frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+…+\frac{1}{\sqrt{99}+\sqrt{100}}.\]

\[=\frac{\sqrt{2}-\sqrt{1}}{\left( \sqrt{2}+\sqrt{1} \right)\left( \sqrt{2}-\sqrt{1} \right)}+\frac{\sqrt{3}-\sqrt{2}}{\left( \sqrt{3}+\sqrt{2} \right)\left( \sqrt{3}-\sqrt{2} \right)}+…+\frac{\sqrt{100}-\sqrt{99}}{\left( \sqrt{100}+\sqrt{99} \right)\left( \sqrt{100}-\sqrt{99} \right)}\]

\[=\frac{\sqrt{2}-\sqrt{1}}{2-1}+\frac{\sqrt{3}-\sqrt{2}}{3-2}+…+\frac{\sqrt{100}-\sqrt{99}}{100-99}\]

\[=\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+…+\sqrt{100}-\sqrt{99}\]

\[=\sqrt{100}-\sqrt{1}\]

\[=10-1=9.\]

b) \[\frac{1}{\sqrt{1}-\sqrt{2}}-\frac{1}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}-\sqrt{4}}-…-\frac{1}{\sqrt{50}-\sqrt{51}}.\]

\[=\frac{\sqrt{1}+\sqrt{2}}{\left( \sqrt{1}-\sqrt{2} \right)\left( \sqrt{1}+\sqrt{2} \right)}-\frac{\sqrt{2}+\sqrt{3}}{\left( \sqrt{2}-\sqrt{3} \right)\left( \sqrt{2}+\sqrt{3} \right)}+\frac{\sqrt{3}+\sqrt{4}}{\left( \sqrt{3}-\sqrt{4} \right)\left( \sqrt{3}+\sqrt{4} \right)}-…-\frac{\sqrt{50}+\sqrt{51}}{\left( \sqrt{50}-\sqrt{51} \right)\left( \sqrt{50}+\sqrt{51} \right)}\]

\[=\frac{\sqrt{1}+\sqrt{2}}{1-2}-\frac{\sqrt{2}+\sqrt{3}}{2-3}+\frac{\sqrt{3}+\sqrt{4}}{3-4}-…-\frac{\sqrt{50}+\sqrt{51}}{50-51}\]

\[=\frac{\sqrt{1}+\sqrt{2}}{-1}-\frac{\sqrt{2}+\sqrt{3}}{-1}+\frac{\sqrt{3}+\sqrt{4}}{-1}-…-\frac{\sqrt{50}+\sqrt{51}}{-1}\]

\[=-\sqrt{1}-\sqrt{2}+\sqrt{2}+\sqrt{3}-\sqrt{3}-\sqrt{4}+…+\sqrt{50}+\sqrt{51}\]

\[=\sqrt{51}-1.\]

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