Thực hiện phép tính: \({\left( {\sqrt 7  + \sqrt 5 } \right)^5} - {\left( {\sqrt 7  - \sqrt 5 } \right)^5}\).

Ta có: \({\left( {\sqrt 7  + \sqrt 5 } \right)^5} - {\left( {\sqrt 7  - \sqrt 5 } \right)^5}\)

\( = \left[ {{{\left( {\sqrt 7 } \right)}^5} + 5.{{\left( {\sqrt 7 } \right)}^4}.\sqrt 5  + 10.{{\left( {\sqrt 7 } \right)}^3}.{{\left( {\sqrt 5 } \right)}^2} + 10.{{\left( {\sqrt 7 } \right)}^2}.{{\left( {\sqrt 5 } \right)}^3} + 5.\sqrt 7 .{{\left( {\sqrt 5 } \right)}^4} + {{\left( {\sqrt 5 } \right)}^5}} \right]\)

\( - \left[ {{{\left( {\sqrt 7 } \right)}^5} - 5.{{\left( {\sqrt 7 } \right)}^4}.\sqrt 5  + 10.{{\left( {\sqrt 7 } \right)}^3}.{{\left( {\sqrt 5 } \right)}^2} - 10.{{\left( {\sqrt 7 } \right)}^2}.{{\left( {\sqrt 5 } \right)}^3} + 5.\sqrt 7 .{{\left( {\sqrt 5 } \right)}^4} - {{\left( {\sqrt 5 } \right)}^5}} \right]\)

\( = 10.{\left( {\sqrt 7 } \right)^4}.\sqrt 5  + 20.{\left( {\sqrt 7 } \right)^2}.{\left( {\sqrt 5 } \right)^3} + 2.{\left( {\sqrt 5 } \right)^5}\)

\( = 490\sqrt 5  + 700\sqrt 5  + 50\sqrt 5  = 1240\sqrt 5 \).