To solve the expression \( 1024:2^4 + 140:(38+25) - 7^{23}:7^{21} \), we need to follow the order of operations (PEMDAS/BODMAS):

1. **Parentheses/Brackets**: Solve expressions inside parentheses first.
2. **Exponents/Orders**: Solve exponents (powers and roots).
3. **Multiplication and Division**: From left to right.
4. **Addition and Subtraction**: From left to right.

Let's break it down step by step:

1. **Parentheses/Brackets**:
- \( 38 + 25 = 63 \)

So the expression now is:
\[
1024:2^4 + 140:63 - 7^{23}:7^{21}
\]

2. **Exponents/Orders**:
- \( 2^4 = 16 \)
- \( 7^{23}:7^{21} = 7^{23-21} = 7^2 = 49 \)

So the expression now is:
\[
1024:16 + 140:63 - 49
\]

3. **Multiplication and Division**:
- \( 1024:16 = 64 \)
- \( 140:63 \approx 2.2222 \)

So the expression now is:
\[
64 + 2.2222 - 49
\]

4. **Addition and Subtraction**:
- \( 64 + 2.2222 = 66.2222 \)
- \( 66.2222 - 49 = 17.2222 \)

So, the final result of the expression \( 1024:2^4 + 140:(38+25) - 7^{23}:7^{21} \) is approximately \( 17.2222 \).