Find the terms independent of x in the expansion of `(sqrt(x)+(1)/(3x^(2)))^(10)`

Explanation

To find the terms independent of \( x \) in the expansion of \( \left(\sqrt{x} + \frac{1}{3x^2}\right)^{10} \), we can follow these steps:

### Step 1: Identify the General Term
The general term \( T_r \) in the binomial expansion of \( (a + b)^n \) is given by:
\[ T_r = \binom{n}{r} a^{n-r} b^r \]
In our case, \( a = \sqrt{x} \), \( b = \frac{1}{3x^2} \), and \( n = 10 \). Therefore, the general term becomes:
\[ T_r = \binom{10}{r} (\sqrt{x})^{10-r} \left(\frac{1}{3x^2}\right)^r \]

### Step 2: Simplify the General Term
Now, we simplify \( T_r \):
\[ T_r = \binom{10}{r} (\sqrt{x})^{10-r} \cdot \frac{1}{3^r} \cdot \frac{1}{x^{2r}} \]
\[ = \binom{10}{r} \cdot \frac{1}{3^r} \cdot x^{\frac{10-r}{2}} \cdot x^{-2r} \]
\[ = \binom{10}{r} \cdot \frac{1}{3^r} \cdot x^{\frac{10 - r - 4r}{2}} \]
\[ = \binom{10}{r} \cdot \frac{1}{3^r} \cdot x^{\frac{10 - 5r}{2}} \]

### Step 3: Set the Power of \( x \) to Zero
To find the term independent of \( x \), we need to set the exponent of \( x \) to zero:
\[ \frac{10 - 5r}{2} = 0 \]
Multiplying both sides by 2 gives:
\[ 10 - 5r = 0 \]
Solving for \( r \):
\[ 5r = 10 \implies r = 2 \]

### Step 4: Find the Independent Term
Now, we substitute \( r = 2 \) back into the general term to find the independent term:
\[ T_2 = \binom{10}{2} \cdot \frac{1}{3^2} \cdot x^{\frac{10 - 5 \cdot 2}{2}} \]
Calculating \( T_2 \):
\[ T_2 = \binom{10}{2} \cdot \frac{1}{9} \cdot x^0 \]
\[ = \frac{10 \cdot 9}{2 \cdot 1} \cdot \frac{1}{9} \]
\[ = 45 \cdot \frac{1}{9} = 5 \]

### Conclusion
The term independent of \( x \) in the expansion is:
\[ \boxed{5} \]