tính nhanh
a, 75^2 + 65^2 + 55^2 - 45^2 - 35^2 - 25^2   b, (9+10)*(9^2+1)*(9^4+1)*(9^8+1)*(9^16+1)

Let's solve each part of the problem:

### Part (a):
We have the expression:
\[
75^2 + 65^2 + 55^2 - 45^2 - 35^2 - 25^2
\]

To calculate this efficiently, let's apply the identity:
\[
a^2 - b^2 = (a - b)(a + b)
\]

Now, group the terms to simplify:

\[
(75^2 - 45^2) + (65^2 - 35^2) + (55^2 - 25^2)
\]

Applying the difference of squares formula:

\[
(75 - 45)(75 + 45) + (65 - 35)(65 + 35) + (55 - 25)(55 + 25)
\]

Simplifying each:

\[
(30)(120) + (30)(100) + (30)(80)
\]

Now, factor out the common factor \(30\):

\[
30 \times (120 + 100 + 80) = 30 \times 300 = 9000
\]

So, the answer to part (a) is \( \boxed{9000} \).

### Part (b):
We need to calculate:
\[
(9 + 10) \times (9^2 + 1) \times (9^4 + 1) \times (9^8 + 1) \times (9^{16} + 1)
\]

Notice that \(9 + 10 = 19\). 

Now, observe that \(9^2 = 81\), \(9^4 = (9^2)^2 = 81^2\), and so on. The expression can be simplified as:

\[
19 \times (81 + 1) \times (81^2 + 1) \times (81^4 + 1) \times (81^8 + 1)
\]

Let's look at each term:
- \(9^2 + 1 = 81 + 1 = 82\)
- \(9^4 + 1 = 81^2 + 1\)
- \(9^8 + 1 = (81^2)^2 + 1\)
- \(9^{16} + 1 = (81^4)^2 + 1\)

These terms become very large, but multiplying them gives:

\[
19 \times 82 \times \text{(a large product)}
\]

Given that the exact large product is complex, we can recognize that this is a type of expansion that simplifies under modular arithmetic. In particular, the structure often relates to a pattern or modulo simplification. 

However, directly calculating the exact large product isn't straightforward without numerical tools due to its size. Therefore, the best approach here is computational:

\[
19 \times 82 = 1558
\]
\[
\text{Total Product: A large number (in the range of quintillions to undecillions)}
\]

So the answer to part (b) involves evaluating an extremely large number.

Do you need further details or have any questions?  

### Related Questions:
1. How can modular arithmetic help simplify large products?
2. What are the properties of difference of squares in algebraic identities?
3. How does factorization simplify complex expressions?
4. What techniques can be used to approximate large products?
5. How do powers of numbers relate in expansions like \(9^n\)?

### Tip:
When dealing with extremely large products or sums, consider breaking down the problem using algebraic identities or modular arithmetic to simplify the calculations.

Explore how to solve complex math problems involving difference of squares and large product expansions. Learn the step-by-step process to solve specific mathematical expressions. Suitable for advanced high school and college students.

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