Utilizing binomial hypothesis the articulations, $$ \[\left( x\text{ }+\text{ }1 \right)6\] and \[\left( x\text{ }\text{ }1 \right)6\]  can be communicated as

\[\left( x\text{ }+\text{ }1 \right)6\text{ }=\text{ }6C0\text{ }x6\text{ }+\text{ }6C1\text{ }x5\text{ }+\text{ }6C2\text{ }x4\text{ }+\text{ }6C3\text{ }x3\text{ }+\text{ }6C4\text{ }x2\text{ }+\text{ }6C5\text{ }x\text{ }+\text{ }6C6\]

\[\left( x\text{ }\text{ }1 \right)6\text{ }=\text{ }6C0\text{ }x6\text{ }\text{ }6C1\text{ }x5\text{ }+\text{ }6C2\text{ }x4\text{ }\text{ }6C3\text{ }x3\text{ }+\text{ }6C4\text{ }x2\text{ }\text{ }6C5\text{ }x\text{ }+\text{ }6C6\]

Presently, \[\left( x\text{ }+\text{ }1 \right)6\text{ }\text{ }\left( x\text{ }\text{ }1 \right)6\]

\[=\text{ }6C0\text{ }x6\text{ }+\text{ }6C1\text{ }x5\text{ }+\text{ }6C2\text{ }x4\text{ }+\text{ }6C3\text{ }x3\text{ }+\text{ }6C4\text{ }x2\text{ }+\text{ }6C5\text{ }x\text{ }+\text{ }6C6\text{ }\text{ }\left[ 6C0\text{ }x6\text{ }\text{ }6C1\text{ }x5\text{ }+\text{ }6C2\text{ }x4\text{ }\text{ }6C3\text{ }x3\text{ }+\text{ }6C4\text{ }x2\text{ }\text{ }6C5\text{ }x\text{ }+\text{ }6C6 \right]\]

\[=\text{ }2\text{ }\left[ 6C0\text{ }x6\text{ }+\text{ }6C2\text{ }x4\text{ }+\text{ }6C4\text{ }x2\text{ }+\text{ }6C6 \right]\]

\[=\text{ }2\text{ }\left[ x6\text{ }+\text{ }15×4\text{ }+\text{ }15×2\text{ }+\text{ }1 \right]\]

Presently by subbing \[x\text{ }=\text{ }\surd 2\] we get

\[\left( \surd 2\text{ }+\text{ }1 \right)6\text{ }\text{ }\left( \surd 2\text{ }\text{ }1 \right)6\text{ }=\text{ }2\text{ }\left[ \left( \surd 2 \right)6\text{ }+\text{ }15\left( \surd 2 \right)4\text{ }+\text{ }15\left( \surd 2 \right)2\text{ }+\text{ }1 \right]\]

\[=\text{ }2\text{ }\left( 8\text{ }+\text{ }15\text{ }\times \text{ }4\text{ }+\text{ }15\text{ }\times \text{ }2\text{ }+\text{ }1 \right)\]

\[=\text{ }2\text{ }\left( 8\text{ }+\text{ }60\text{ }+\text{ }30\text{ }+\text{ }1 \right)\]

\[=\text{ }2\text{ }\left( 99 \right)\] \[=\text{ }198\]