Rút gọn biểu thức:

a). \[2\sqrt {{x^2}} \], với \[x < 0\];                                                                    b). \[\frac{1}{2}\sqrt {{x^{10}}} \], với \[x < 0\];

c). \[\sqrt {{{\left( {a - 5} \right)}^2}} \], với \[a \le 5\];                                                                   d). \[\sqrt {{{\left( {x - 10} \right)}^{10}}} \], với \[x \le 10\];

e). \[x - 4 + \sqrt {{x^2} - 8x + 16} \], với \[x < 4\];                                                                   f). \[\sqrt {{{\left( {\sqrt x  - \sqrt y } \right)}^2}{{\left( {\sqrt x  - \sqrt y } \right)}^2}} \], với \[0 \le x \le y.\]

a) \[2\sqrt {{x^2}}  = 2\left| x \right| =  - 2x.\]

b) \[\frac{1}{2}\sqrt {{x^{10}}}  = \frac{1}{2}\sqrt {{x^{5.2}}}  = \frac{1}{2}\sqrt {{{\left( {{x^5}} \right)}^2}}  = \frac{1}{2}\left| {{x^5}} \right| =  - \frac{1}{2}{x^5}.\]

c) Ta có:

• \[a \le 5 \Rightarrow a - 5 \le 0\]     • \[\sqrt {{{\left( {a - 5} \right)}^2}}  = \left| {a - 5} \right| = 5 - a\]

d) Ta có:

• \[x \le 10 \Rightarrow x - 10 \le 0 \Rightarrow {\left( {x - 10} \right)^5} \le 0 \Rightarrow {\left( {10 - x} \right)^5} \ge 0.\]

• \[\sqrt {{{\left( {x - 10} \right)}^{10}}}  = \sqrt {{{\left[ {{{\left( {x - 10} \right)}^5}} \right]}^2}}  = \left| {{{\left( {x - 10} \right)}^5}} \right| = \left| {{{\left( {10 - x} \right)}^5}} \right| = {\left( {10 - x} \right)^5}.\]

e) Ta có:

• \[x < 4 \Rightarrow x - 4 < 0\]

• \[x - 4 + \sqrt {{x^2} - 8x + 16}  = x - 4 + \sqrt {{{\left( {x - 4} \right)}^2}}  = x - 4 + \left| {x - 4} \right| = x - 4 - \left( {x - 4} \right) = 0\]

f) Ta có:

• \[0 \le x \le y\] nên \[x - y \le 0\] hay \[y - x \ge 0\]

• \[\sqrt {{{\left( {\sqrt x  - \sqrt y } \right)}^2}{{\left( {\sqrt x  + \sqrt y } \right)}^2}}  = \sqrt {{{\left[ {\left( {\sqrt x  - \sqrt y } \right)\left( {\sqrt x  + \sqrt y } \right)} \right]}^2}}  = \sqrt {{{\left[ {\sqrt {{x^2}}  - \sqrt {{y^2}} } \right]}^2}} \]

\[ = \sqrt {{{\left( {x - y} \right)}^2}}  = \left| {x - y} \right| =  - \left( {x - y} \right) = y - x.\]