Khai triển \({\left( {x + \frac{1}{x}} \right)^4}\). Khi đó:

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

\[ = 243 + 405\sqrt 2  + 540 + 180\sqrt 2  + 60 + 4\sqrt 2 \]\[ = 843 + 589\sqrt 2 \].

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

\[ = 243 - 405\sqrt 2  + 540 - 180\sqrt 2  + 60 - 4\sqrt 2 \]\[ = 843 - 589\sqrt 2 \].

Vậy \({\left( {3 + \sqrt 2 } \right)^5} - {\left( {3 - \sqrt 2 } \right)^5}\)\[ = 843 + 589\sqrt 2  - \left( {843 - 589\sqrt 2 } \right) = 1178\sqrt 2 \].

Suy ra \(a = 0;b = 1178\). Vậy \(a + b = 1178\).

Trả lời: 1178.

\({\left( {x + \frac{1}{{2x}}} \right)^4} = {x^4} + 4 \cdot {x^3} \cdot \left( {\frac{1}{{2x}}} \right) + 6 \cdot {x^2} \cdot {\left( {\frac{1}{{2x}}} \right)^2} + 4 \cdot x \cdot {\left( {\frac{1}{{2x}}} \right)^3} + {\left( {\frac{1}{{2x}}} \right)^4}\)\( = {x^4} + 2{x^2} + \frac{3}{2} + \frac{1}{{2{x^2}}} + \frac{1}{{16{x^4}}}\).

Số hạng không chứa \(x\) là \(\frac{3}{2} = 1,5\).

Trả lời: 1,5.