Expand the binomial `((x^(2))/(3) + (3)/(x))^(5)`

Answer

To expand the binomial expression \(\left(\frac{x^2}{3} + \frac{3}{x}\right)^5\), we will use the Binomial Theorem, which states that:

\[ (a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \]

In our case, let:
- \(a = \frac{x^2}{3}\)
- \(b = \frac{3}{x}\)
- \(n = 5\)

Now, we can expand the expression step by step.

### Step 1: Identify the terms
We have:
\[ \left(\frac{x^2}{3} + \frac{3}{x}\right)^5 \]

### Step 2: Apply the Binomial Theorem
Using the Binomial Theorem, we can write:
\[ \left(\frac{x^2}{3} + \frac{3}{x}\right)^5 = \sum_{k=0}^{5} \binom{5}{k} \left(\frac{x^2}{3}\right)^{5-k} \left(\frac{3}{x}\right)^k \]

### Step 3: Calculate each term in the summation
We will calculate each term for \(k = 0\) to \(k = 5\):

1. **For \(k = 0\)**:
\[ \binom{5}{0} \left(\frac{x^2}{3}\right)^5 \left(\frac{3}{x}\right)^0 = 1 \cdot \frac{x^{10}}{243} = \frac{x^{10}}{243} \]

2. **For \(k = 1\)**:
\[ \binom{5}{1} \left(\frac{x^2}{3}\right)^4 \left(\frac{3}{x}\right)^1 = 5 \cdot \frac{x^8}{81} \cdot \frac{3}{x} = \frac{15x^8}{81} = \frac{5x^8}{27} \]

3. **For \(k = 2\)**:
\[ \binom{5}{2} \left(\frac{x^2}{3}\right)^3 \left(\frac{3}{x}\right)^2 = 10 \cdot \frac{x^6}{27} \cdot \frac{9}{x^2} = \frac{90x^6}{27} = \frac{10x^6}{3} \]

4. **For \(k = 3\)**:
\[ \binom{5}{3} \left(\frac{x^2}{3}\right)^2 \left(\frac{3}{x}\right)^3 = 10 \cdot \frac{x^4}{9} \cdot \frac{27}{x^3} = \frac{270x^4}{9} = 30x^4 \]

5. **For \(k = 4\)**:
\[ \binom{5}{4} \left(\frac{x^2}{3}\right)^1 \left(\frac{3}{x}\right)^4 = 5 \cdot \frac{x^2}{3} \cdot \frac{81}{x^4} = \frac{405}{3x^2} = \frac{135}{x^2} \]

6. **For \(k = 5\)**:
\[ \binom{5}{5} \left(\frac{x^2}{3}\right)^0 \left(\frac{3}{x}\right)^5 = 1 \cdot 1 \cdot \frac{243}{x^5} = \frac{243}{x^5} \]

### Step 4: Combine all terms
Now, we combine all the terms we calculated:
\[ \frac{x^{10}}{243} + \frac{5x^8}{27} + \frac{10x^6}{3} + 30x^4 + \frac{135}{x^2} + \frac{243}{x^5} \]

### Final Answer
Thus, the expansion of \(\left(\frac{x^2}{3} + \frac{3}{x}\right)^5\) is:
\[ \frac{x^{10}}{243} + \frac{5x^8}{27} + \frac{10x^6}{3} + 30x^4 + \frac{135}{x^2} + \frac{243}{x^5} \]