by a) Tính: \(\sqrt {{{\left( {2 – \sqrt 5 } \right)}^2}} – \sqrt {\frac{8}{{7 – 3\sqrt 5 }}} \)
b) Rút gọn: \(A = \frac{x}{{\sqrt x – 1}} – \frac{{2x – \sqrt x }}{{x – \sqrt x }}\) (với \(x > 0{;^{}}x \ne 1\))
A 3
B 4
C -5
D 7
Hướng dẫn Chọn đáp án là: C
Lời giải chi tiết:
\(\begin{array}{l}
\sqrt {{{\left( {2 – \sqrt 5 } \right)}^2}} – \sqrt {\frac{8}{{7 – 3\sqrt 5 }}} \\
= \left| {2 – \sqrt 5 } \right| – \sqrt {\frac{{16}}{{14 – 6\sqrt 5 }}} \\
= \left| {2 – \sqrt 5 } \right| – \frac{{\sqrt {16} }}{{\sqrt {14 – 6\sqrt 5 } }}\\
= – \left( {2 – \sqrt 5 } \right) – \frac{4}{{\sqrt {{3^2} – 2.3.\sqrt 5 + {{\sqrt 5 }^2}} }}\left( {Do{\rm{ }}2 – \sqrt 5 < 0} \right)\\
= – 2 + \sqrt 5 – \frac{4}{{\sqrt {{{\left( {3 – \sqrt 5 } \right)}^2}} }}\\
= – 2 + \sqrt 5 – \frac{4}{{3 – \sqrt 5 }}\\
= – 2 + \sqrt 5 – \frac{{4.\left( {3 + \sqrt 5 } \right)}}{{\left( {3 – \sqrt 5 } \right)\left( {3 + \sqrt 5 } \right)}}\\
= – 2 + \sqrt 5 – \frac{{4.\left( {3 + \sqrt 5 } \right)}}{4}\\
= – 2 + \sqrt 5 – 3 – \sqrt 5 = – 5
\end{array}\)
b)
\(\begin{array}{l}
A = \frac{x}{{\sqrt x – 1}} – \frac{{2x – \sqrt x }}{{x – \sqrt x }}\\
= \frac{x}{{\sqrt x – 1}} – \frac{{\sqrt x \left( {2\sqrt x – 1} \right)}}{{\sqrt x \left( {\sqrt x – 1} \right)}}\\
= \frac{x}{{\sqrt x – 1}} – \frac{{2\sqrt x – 1}}{{\sqrt x – 1}}\\
= \frac{{x – 2\sqrt x + 1}}{{\sqrt x – 1}}\\
= \frac{{{{\left( {\sqrt x – 1} \right)}^2}}}{{\sqrt x – 1}} = \sqrt x – 1
\end{array}\)
