Rút gọn các biểu thức :
a) $A = C^1_{2n}+C^3_{2n}+...+C^{2n-1}_{2n}$
b) $B = C^0_{2n}+C^2_{2n}+...+C^{2n}_{2n}$
Ta có :
      $(1+x)^{2n} = C^0_{2n}+C^1_{2n}x+...+C^{2n-1}_{2n}x^{2n-1}+C^{2n}_{2n}x^{2n}.$        (1)
      $(1-x)^{2n} = C^0_{2n}-C^1_{2n}x+...-C^{2n-1}_{2n}x^{2n-1}+C^{2n}_{2n}x^{2n}.$           (2)
Thay $x=1$ vào (1) và (2) ta được :
      $2^{2n} = C^0_{2n}+C^1_{2n}+...+C^{2n-1}_{2n}+C^{2n}_{2n}.$                                         (3)
      $0 = C^0_{2n}-C^1_{2n}+...-C^{2n-1}_{2n}+C^{2n}_{2n}.$                                                   (4)
a) Trừ (1) cho (2) vế theo vế ta được :
      $2(C^1_{2n}+C^3_{2n}+...+C^{2n-1}_{2n}) = 2^{2n} \Leftrightarrow  A = 2^{2n-1}.$
b) Cộng (1) cho (2) vế theo vế ta được :
      $2(C^0_{2n}+C^2_{2n}+...+C^{2n}_{2n}) = 2^{2n} \Leftrightarrow  B = 2^{2n-1}.$
1
phiếu
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Chứng minh :
$\frac{{\sqrt[3]{{a + \sqrt {2 - {a^2}} }}.\sqrt[6]{{1 - a\sqrt {2 - {a^2}}
}}}}{{\sqrt[3]{{1 - {a^2}}}}} =\begin{cases}\sqrt[6]{2}\,\,\,\,\,nếu\,\,\,\,\,\left| a \right| < 1 \\ - \sqrt[6]{2}\,\,\,\,\,\,nếu\,\,\,\,\,1 < \left| a \right| \le 2 \end{cases} $
0
phiếu
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Rút gọn
\(
B= \left ( \frac{1}{a}+\frac{1}{b}-\frac{2c}{ab} \right )\left ( a+b+2c \right )\div  \left ( \frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{2}{ab}-\frac{4c^{2}}{a^{2}b^{2}}\right )
\)
0
phiếu
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Rút gọn
\(
A=a - [\frac{\left ( 16-a \right )a}{a^{2}-4}+\frac{3+2a}{2-a}+\frac{3a-2}{2+a}] \div  \frac{a-1}{a\left ( a^{2} +4a+4\right )}
\)
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phiếu
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Rút gọn biểu thức
\(
C=[\left (\frac{a+1}{a-1} \right )^{2}+3]\div [ \left ( \frac{a-1}{a+1} \right )^{2}+3 ] \div  \frac{a^{3}+1}{a^{3}-1} - \frac{2a}{a-1}
\)
0
phiếu
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Rút gọn biểu thức:
\( B=\frac{\frac{1}{a}+\frac{1}{b+c}}{\frac{1}{a}-\frac{1}{b+c}}\times \left ( 1+\frac{b^{2}+c^{2}-a^{2}}{2ab} \right )\times \left ( a+b+c \right )^{-2} \)

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