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You’ve shared two binomial expansion problems. Let's solve them step-by-step:

### Problem 9: Fifth term of \((2x - 5y)^6\)
We can use the binomial expansion formula:
\[
T_{k+1} = \binom{n}{k} a^{n-k} b^k
\]
where \(n\) is the power of the binomial, \(a\) and \(b\) are the terms in the binomial, and \(k\) is the term number minus 1.

For the fifth term, \(k = 4\), \(n = 6\), \(a = 2x\), and \(b = -5y\). The general term for the expansion of \((2x - 5y)^6\) is:

\[
T_{k+1} = \binom{6}{4} (2x)^{6-4} (-5y)^4
\]

Now calculate:
1. \(\binom{6}{4} = \frac{6!}{4!(6-4)!} = 15\)
2. \((2x)^2 = 4x^2\)
3. \((-5y)^4 = 625y^4\)

Now, multiply everything together:
\[
T_5 = 15 \cdot 4x^2 \cdot 625y^4 = 15 \cdot 2500x^2y^4 = 37500x^2y^4
\]

Thus, the fifth term is **\(37500x^2y^4\)**.

### Problem 10: Eighth term of \((3x + y)^{16}\)
Similarly, we apply the binomial expansion formula for this problem. Here, \(n = 16\), \(a = 3x\), and \(b = y\), and we want the eighth term, so \(k = 7\).

The general term is:
\[
T_{k+1} = \binom{16}{7} (3x)^{16-7} y^7
\]

Now calculate:
1. \(\binom{16}{7} = 11440\)
2. \((3x)^9 = 19683x^9\)
3. \(y^7 = y^7\)

Now, multiply everything together:
\[
T_8 = 11440 \cdot 19683x^9 \cdot y^7 = 225118320x^9y^7
\]

Thus, the eighth term is **\(225118320x^9y^7\)**.

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Let me know if you want further clarifications or details on these steps!

### Related Questions:
1. What is the third term of \((2x - 5y)^6\)?
2. Can you find the fourth term of \((3x + y)^{16}\)?
3. How would you find the seventh term of \((2x - 5y)^8\)?
4. What is the general form for the binomial expansion of \((a + b)^n\)?
5. How do binomial coefficients relate to Pascal’s triangle?

**Tip:** Binomial expansions can always be simplified by breaking them down term by term. Each term follows the pattern based on the binomial theorem!

Write the fifth term of (2x - 5y)^6 and the eighth term of (3x + y)^16.

This page explains how to find the fifth term of (2x - 5y)^6 and the eighth term of (3x + y)^16 using the binomial expansion formula. Detailed step-by-step calculations are provided for binomial coefficients and the general term.

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