Find the term independent of x in
`(1+3x)^(n)(1+(1)/(3x))^(n)`.

Answer

To find the term independent of \( x \) in the expression \( (1 + 3x)^n \left(1 + \frac{1}{3x}\right)^n \), we can follow these steps:

### Step 1: Rewrite the Expression
We start with the expression:
\[ (1 + 3x)^n \left(1 + \frac{1}{3x}\right)^n \]
This can be rewritten as:
\[ (1 + 3x)^n \cdot \left(1 + \frac{1}{3x}\right)^n = (1 + 3x)^n \cdot \left(\frac{3x + 1}{3x}\right)^n = (1 + 3x)^n \cdot \left(1 + 3x\right)^n \cdot \left(\frac{1}{3x}\right)^n \]

### Step 2: Combine the Terms
Now, we can combine the two binomial expansions:
\[ (1 + 3x)^n \cdot (1 + 3x)^n = (1 + 3x)^{2n} \]
Thus, we have:
\[ (1 + 3x)^{2n} \cdot \left(\frac{1}{3x}\right)^n = \frac{(1 + 3x)^{2n}}{(3x)^n} \]

### Step 3: Expand the Binomial
Next, we expand \( (1 + 3x)^{2n} \) using the binomial theorem:
\[ (1 + 3x)^{2n} = \sum_{r=0}^{2n} \binom{2n}{r} (3x)^r \]
This gives us:
\[ = \sum_{r=0}^{2n} \binom{2n}{r} 3^r x^r \]

### Step 4: Substitute into the Expression
Now substituting back into our expression, we have:
\[ \frac{1}{(3x)^n} \sum_{r=0}^{2n} \binom{2n}{r} 3^r x^r = \sum_{r=0}^{2n} \binom{2n}{r} 3^{r-n} x^{r-n} \]

### Step 5: Find the Term Independent of \( x \)
To find the term independent of \( x \), we need \( r - n = 0 \) or \( r = n \). Therefore, we need to find the coefficient when \( r = n \):
\[ \text{Term independent of } x = \binom{2n}{n} 3^{n-n} = \binom{2n}{n} \]

### Final Result
Thus, the term independent of \( x \) in the expression \( (1 + 3x)^n \left(1 + \frac{1}{3x}\right)^n \) is:
\[ \boxed{\binom{2n}{n}} \]