Write down and simplify
Find the 4th term the end in `(2a + 5b)^8`

Answer

To find the 4th term from the end of the expression \((2a + 5b)^8\), we can use the Binomial Theorem, which states that:

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

### Step-by-Step Solution:

1. **Identify the terms**: In our case, \(x = 2a\), \(y = 5b\), and \(n = 8\).

2. **Find the total number of terms**: The total number of terms in the expansion of \((x + y)^n\) is \(n + 1\). Here, \(n + 1 = 8 + 1 = 9\).

3. **Determine the position of the 4th term from the end**: The 4th term from the end corresponds to the \(9 - 4 = 5\)th term from the beginning.

4. **Use the Binomial Theorem to find the 5th term**: The \(r\)th term in the expansion is given by:

\[ T_{r+1} = \binom{n}{r} x^{n-r} y^r \]

For the 5th term, we have \(r = 4\) (since \(r\) starts from 0):

\[ T_5 = \binom{8}{4} (2a)^{8-4} (5b)^4 \]

5. **Calculate the binomial coefficient**:

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

6. **Calculate the powers**:

\[ (2a)^{4} = 16a^4 \quad \text{and} \quad (5b)^{4} = 625b^4 \]

7. **Combine the results**:

\[ T_5 = 70 \cdot 16a^4 \cdot 625b^4 \]

8. **Multiply the coefficients**:

\[ 70 \cdot 16 \cdot 625 = 700000 \]

9. **Final expression**:

\[ T_5 = 700000 a^4 b^4 \]

### Conclusion:
The 4th term from the end of \((2a + 5b)^8\) is:

\[ 700000 a^4 b^4 \]