Find the term that is independent of x in the expansion of the following(x^2 + 1/2x^6)^16

We are tasked with finding the term that is independent of \( x \) in the expansion of \( \left( x^2 + \frac{1}{2x^6} \right)^{16} \).

To solve this, we'll use the binomial expansion formula:
\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]
In this case:
- \( a = x^2 \)
- \( b = \frac{1}{2x^6} \)
- \( n = 16 \)

The general term of the expansion is given by:
\[
T_k = \binom{16}{k} (x^2)^{16-k} \left( \frac{1}{2x^6} \right)^k
\]
Simplifying the powers of \( x \) in the general term:

\[
T_k = \binom{16}{k} x^{2(16-k)} \cdot \frac{1}{2^k x^{6k}} = \binom{16}{k} \frac{1}{2^k} x^{32 - 8k}
\]

For the term to be independent of \( x \), the power of \( x \) must be zero. Hence, we need to solve:
\[
32 - 8k = 0
\]
\[
k = 4
\]

Substituting \( k = 4 \) into the general term to find the constant term:
\[
T_4 = \binom{16}{4} \frac{1}{2^4} x^0 = \binom{16}{4} \frac{1}{16}
\]

Now, calculate \( \binom{16}{4} \):
\[
\binom{16}{4} = \frac{16 \times 15 \times 14 \times 13}{4 \times 3 \times 2 \times 1} = 1820
\]

Thus, the constant term is:
\[
T_4 = \frac{1820}{16} = 113.75
\]

So, the term that is independent of \( x \) is \( 113.75 \).

Would you like any details on the binomial expansion or how powers were simplified?

Here are 5 related questions to expand your understanding:
1. What is the general binomial expansion formula and how is it applied in different cases?
2. How would the result change if the exponent was different, such as \( (x^2 + \frac{1}{2x^6})^{12} \)?
3. How do binomial coefficients relate to Pascal's triangle?
4. Can you explain how negative exponents affect the binomial expansion?
5. What happens if fractional exponents are involved in the binomial expansion?

**Tip**: Always check powers of variables carefully when simplifying terms to ensure accuracy in expansions.

Find the term that is independent of x in the expansion of the following (x^2 + 1/2x^6)^16

Learn how to find the term that is independent of x in the binomial expansion of (x^2 + 1/2x^6)^16. This detailed solution walks through the binomial expansion formula, algebraic simplifications, and calculating the constant term using combinations. Suitable for high school students in Grades 10-12.

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