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The problem asks to find the coefficient independent of \(x\) in the expansion of:

\[
\left( x^3 + \frac{2}{x} \right)^{20}
\]

### Approach:

We need to use the binomial theorem for this expression. The general term in the binomial expansion of \((x^3 + \frac{2}{x})^{20}\) is given by:

\[
T_r = \binom{20}{r} (x^3)^{20-r} \left( \frac{2}{x} \right)^r
\]

Simplifying this general term:

\[
T_r = \binom{20}{r} x^{3(20-r)} \cdot \frac{2^r}{x^r}
\]

\[
T_r = \binom{20}{r} 2^r x^{60 - 4r}
\]

For the term to be independent of \(x\), the exponent of \(x\) must be 0. Therefore, we solve for \(r\) such that:

\[
60 - 4r = 0
\]

\[
r = 15
\]

### Now, we substitute \(r = 15\) into the general term to find the coefficient:

\[
T_{15} = \binom{20}{15} 2^{15} x^0
\]

The binomial coefficient \(\binom{20}{15} = \binom{20}{5}\) (since \(\binom{n}{k} = \binom{n}{n-k}\)):

\[
\binom{20}{5} = \frac{20 \times 19 \times 18 \times 17 \times 16}{5 \times 4 \times 3 \times 2 \times 1} = 15504
\]

Thus, the coefficient is:

\[
15504 \times 2^{15} = 15504 \times 32768 = 507805696
\]

### Final Answer:
The coefficient independent of \(x\) is **507,805,696**.

Would you like more details or clarifications?

Here are five related questions for further practice:
1. How does the binomial theorem apply to expressions with fractional exponents?
2. Can you find the term containing \(x^3\) in the expansion?
3. What would be the result if the power in the binomial expansion were increased to 25?
4. How would the method change if the expression were \(\left( x^4 + \frac{3}{x} \right)^{15}\)?
5. What happens to the expansion if both terms in the binomial were negative?

**Tip**: The binomial theorem is particularly powerful for finding specific terms without expanding the entire expression!

Find the coefficient independent of x in (x^3 + 2/x)^20

Solve the problem of finding the coefficient independent of x in the expansion of (x^3 + 2/x)^20 using binomial theorem and exponent balancing. Suitable for students in Grades 10-12, this solution covers binomial expansion, exponent rules, and combinatorics.

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