Rút gọn biểu thức
a)A=3√12-4√3 +5√27
b)B= $\frac{1}{\sqrt{7+4\sqrt{3} }}$
c) C=( $\frac{√x+1}{√x-1}$ – $\frac{x+√x}{x-1}$ ) : ($\frac{1}{√x+1}$ + $\frac{1}{√x-1}$ )(với x >0,x$\neq$ 1)
Rút gọn biểu thức
a)A=3√12-4√3 +5√27
b)B= $\frac{1}{\sqrt{7+4\sqrt{3} }}$
c) C=( $\frac{√x+1}{√x-1}$ – $\frac{x+√x}{x-1}$ ) : ($\frac{1}{√x+1}$ + $\frac{1}{√x-1}$ )(với x >0,x$\neq$ 1)
Đáp án:
$\begin{array}{l}
A = 3\sqrt {12} – 4\sqrt 3 + 5\sqrt {27} \\
= 3.\sqrt {{2^2}.3} – 4\sqrt 3 + 5.\sqrt {{3^2}.3} \\
= 3.2\sqrt 3 – 4\sqrt 3 + 5.3\sqrt 3 \\
= 6\sqrt 3 – 4\sqrt 3 + 15\sqrt 3 \\
= 13\sqrt 3 \\
B = \frac{1}{{\sqrt {7 + 4\sqrt 3 } }} = \frac{1}{{\sqrt {4 + 2.2\sqrt 3 + 3} }}\\
= \frac{1}{{\sqrt {{{\left( {2 + \sqrt 3 } \right)}^2}} }}\\
= \frac{1}{{2 + \sqrt 3 }}\\
= \frac{{2 – \sqrt 3 }}{{\left( {2 + \sqrt 3 } \right)\left( {2 – \sqrt 3 } \right)}}\\
= \frac{{2 – \sqrt 3 }}{{4 – 3}}\\
= 2 – \sqrt 3 \\
C = \left( {\frac{{\sqrt x + 1}}{{\sqrt x – 1}} – \frac{{x + \sqrt x }}{{x – 1}}} \right):\left( {\frac{1}{{\sqrt x + 1}} + \frac{1}{{\sqrt x – 1}}} \right)\left( {x > 0;x \ne 1} \right)\\
= \left[ {\frac{{\sqrt x + 1}}{{\sqrt x – 1}} – \frac{{\sqrt x \left( {\sqrt x + 1} \right)}}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 1} \right)}}} \right]:\frac{{\sqrt x – 1 + \sqrt x + 1}}{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 1} \right)}}\\
= \frac{{\sqrt x + 1}}{{\sqrt x – 1}} – \frac{{\sqrt x }}{{\sqrt x – 1}}.\frac{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 1} \right)}}{{2\sqrt x }}\\
= \frac{1}{{\sqrt x – 1}}.\frac{{\left( {\sqrt x – 1} \right)\left( {\sqrt x + 1} \right)}}{{2\sqrt x }}\\
= \frac{{\sqrt x + 1}}{{2\sqrt x }}
\end{array}$