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

Answer

To expand the binomial \(\left(\frac{x^2}{3} + \frac{3}{x}\right)^5\), we can 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, \(a = \frac{x^2}{3}\), \(b = \frac{3}{x}\), and \(n = 5\).

### Step 1: Identify the terms
We will identify \(a\), \(b\), and \(n\):
- \(a = \frac{x^2}{3}\)
- \(b = \frac{3}{x}\)
- \(n = 5\)

### Step 2: Write the binomial expansion
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: Expand the summation
Now we will expand the summation term by term:

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 the 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:

\[ \frac{x^{10}}{243} + \frac{5x^8}{27} + \frac{10x^6}{3} + 30x^4 + \frac{135}{x^2} + \frac{243}{x^5} \]