Chưng minh dăng thưc:

a) \(\frac{{\sqrt {{a^2} + {x^2}}  + \sqrt {{a^2} - {x^2}} }}{{\sqrt {{a^2} + {x^2}}  - \sqrt {{a^2} - {x^2}} }} - \sqrt {\frac{{{a^4}}}{{{x^4}}} - 1}  = \frac{{{a^2}}}{{{x^2}}};\) với \(|a| > |x|\)

b) \({\left( {\frac{{5 + 2\sqrt 6 }}{{\sqrt 3  + \sqrt 2 }}} \right)^2} - {\left( {\frac{{5 - 2\sqrt 6 }}{{\sqrt 3  - \sqrt 2 }}} \right)^2} = 4\sqrt 6 \)

c) \(\left( {\frac{{x\sqrt x  - y\sqrt y }}{{\sqrt x  - \sqrt y }} + \sqrt {xy} } \right):{(\sqrt x  + \sqrt y )^2} = 1\quad (x > 0,y > 0,x \ge y)\)

a) Ta có \(\sqrt {50}  - \sqrt {32}  + 3\sqrt 8  = \sqrt {25.2}  - \sqrt {16.2}  + 3\sqrt {4.2}  = 5\sqrt 2  - 4\sqrt 2  + 3.2.\sqrt 2  = 7\sqrt 2 \).

b) \(\sqrt {25a}  + 2\sqrt {160a}  - 3\sqrt {10a}  = \sqrt {25.10a}  + 2.\sqrt {16.10a}  - 3\sqrt {10a} \)

\( = 5\sqrt {10a}  + 2.4.\sqrt {10a}  - 3\sqrt {10a}  = 10\sqrt {10a} \).

c) \(\left( {2\sqrt 7  + \sqrt 3 } \right)\sqrt 7  - \sqrt {84}  = 2\sqrt 7 .\sqrt 7  + \sqrt 3 .\sqrt 7  - \sqrt {4.21} \)

\( = 2.7 + \sqrt {21}  - 2\sqrt {21}  = 14 - \sqrt {21} \).

d) \(\left( {\sqrt {63}  - \sqrt 8  - \sqrt 7 } \right)\sqrt 7  + 2\sqrt {14}  = \sqrt {63} .\sqrt 7  - \sqrt 8 .\sqrt 7  - \sqrt 7 .\sqrt 7  + 2\sqrt {14} \)

\( = \sqrt {9.7.7}  - 2\sqrt 2 .\sqrt 7  - 7 + 2\sqrt {14}  = 3.7 - 2\sqrt {14}  - 7 + 2\sqrt {14}  = 14\).

a). Với \(a \ge 0\), ta có: \(3\sqrt {2a}  - \sqrt {18{a^3}}  + 4\sqrt {\frac{a}{2}}  - \frac{1}{4}\sqrt {128a} \)

\( = 3\sqrt {2a}  - 3a\sqrt {2a}  + 2\sqrt {2a}  - 2\sqrt {2a} \)

\( = 3\sqrt {2a}  - 3a\sqrt {2a}  = 3\left( {1 - a} \right)\sqrt {2a} .\)

b). Với \(x > y > 0\), ta có: \(2y\sqrt {x - y}  + x\sqrt {\frac{1}{{x - y}}}  - x\sqrt {\frac{a}{{ax - ay}}}  - \sqrt {{x^3} - {x^2}y} \)

\( = 2y\sqrt {x - y}  + x\sqrt {\frac{1}{{x - y}}}  - x\sqrt {\frac{a}{{a\left( {x - y} \right)}}}  - \sqrt {{x^2}\left( {x - y} \right)} \)

\( = 2y\sqrt {x - y}  - \left| x \right|\sqrt {x - y}  + x\sqrt {\frac{1}{{x - y}}}  - x\sqrt {\frac{1}{{x - y}}} \)

\( = \sqrt {x - y} \left( {2y - x} \right)\) (do \(x > 0\)).

c). Với \(a \ge 0,b \ge 0,a \ne b\), ta có:
\(\frac{{\sqrt a  + \sqrt b }}{{\sqrt a  - \sqrt b }} + \frac{{\sqrt a  - \sqrt b }}{{\sqrt a  + \sqrt b }} = \frac{{{{\left( {\sqrt a  + \sqrt b } \right)}^2} + {{\left( {\sqrt a  - \sqrt b } \right)}^2}}}{{\left( {\sqrt a  - \sqrt b } \right)\left( {\sqrt a  + \sqrt b } \right)}}\)

\( = \frac{{a + 2\sqrt {ab}  + b + a - 2\sqrt {ab}  + b}}{{\left( {\sqrt a  - \sqrt b } \right)\left( {\sqrt a  + \sqrt b } \right)}} = \frac{{2a + 2b}}{{a - b}}\).