To find the term independent of x in the expansion of \(\left( x + \frac{3}{x} \right)^4\), we use the binomial theorem:

\(\left( x + \frac{3}{x} \right)^4 = \sum_{r=0}^{4} \binom{4}{r} x^{4-r} \left( \frac{3}{x} \right)^r\)

The general term is:

\(T_r = \binom{4}{r} x^{4-r} \left( \frac{3}{x} \right)^r = \binom{4}{r} x^{4-r} \cdot \frac{3^r}{x^r} = \binom{4}{r} \cdot 3^r \cdot x^{4-2r}\)

We need the exponent of x to be zero for the term to be independent of x:

\(4 - 2r = 0\)

Solving for r gives:

\(4 = 2r\)

\(r = 2\)

Substitute r back into the general term:

\(T_2 = \binom{4}{2} \cdot 3^2 \cdot x^{4-4} = \binom{4}{2} \cdot 9\)

\(\binom{4}{2} = 6\)

\(T_2 = 6 \cdot 9 = 54\)

Thus, the value of the term independent of x is 54.