find the 5th term in the expansion of (x + y)8

To find the 5th term in the expansion of \((x + y)^8\), we use the **binomial theorem**, which states that:

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

Here, \(n = 8\). The general term in the expansion is:

\[
\binom{8}{k} x^{8-k} y^k
\]

The 5th term corresponds to \(k = 4\), since the terms are indexed starting from \(k = 0\). Substituting \(k = 4\) into the general term:

\[
\text{5th term} = \binom{8}{4} x^{8-4} y^4 = \binom{8}{4} x^4 y^4
\]

Now, calculate \(\binom{8}{4}\):

\[
\binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8 \times 7 \times 6 \times 5}{4 \times 3 \times 2 \times 1} = 70
\]

So, the 5th term is:

\[
70 x^4 y^4
\]

Thus, the 5th term in the expansion of \((x + y)^8\) is **70 \(x^4 y^4\)**.

Would you like more details on any part of this process?

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Here are five related questions for further exploration:
1. How would you find the 6th term in the expansion of \((x + y)^{10}\)?
2. What is the middle term in the expansion of \((x + y)^{9}\)?
3. How does the binomial theorem change if one term has a coefficient, like \((3x + y)^8\)?
4. Can the binomial theorem be applied to negative exponents?
5. What is the relationship between binomial coefficients and Pascal's Triangle?

**Tip:** The binomial coefficients can be quickly found using Pascal's Triangle, which is a simple and visual way to calculate combinations.

find the 5th term in the expansion of (x + y)^8

Learn how to find the 5th term in the binomial expansion of (x + y)^8 using the binomial theorem and combination formula. This step-by-step guide explains the calculation and provides insights into key algebraic concepts. Suitable for students in Grades 10-12.

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