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Let's break down the exercises one by one.

### 7. Aplicando productos notables, calcular el valor de:
\[
16 \times 3 \times 5 \times 17 \times 257 + 1
\]
This is a product of several numbers followed by adding 1. Calculating this product:
\[
16 \times 3 \times 5 \times 17 \times 257 = 1045440
\]
Thus, adding 1:
\[
1045440 + 1 = 1045441
\]

Now, we proceed to the next part of the problem:
\[
\left( \sqrt[3]{3 + 2\sqrt{2} + 3\sqrt[3]{3 - 2\sqrt{2}}}\right)^3 - 9\left( \sqrt[3]{3 + 2\sqrt{2} + 3\sqrt[3]{3 - 2\sqrt{2}}}\right)
\]
Let's call \( x = \sqrt[3]{3 + 2\sqrt{2} + 3\sqrt[3]{3 - 2\sqrt{2}}} \). The expression simplifies to:
\[
x^3 - 9x
\]
Since \( x^3 = 9 \), we substitute and get:
\[
9 - 9x = 9 - 9 = 0
\]

Thus, the final value for exercise 7 is:
\[
1045441 + 0 = 1045441
\]

### 8. Hallar "K", si:
\[
(a + 3b)^2 + (a + b)^2 + Kb^2
\]
This is a perfect square trinomial (es un TCP). Let's expand and simplify the equation to find \( K \).

First, expand both terms:
\[
(a + 3b)^2 = a^2 + 6ab + 9b^2
\]
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
Now sum the expressions:
\[
(a + 3b)^2 + (a + b)^2 = a^2 + 6ab + 9b^2 + a^2 + 2ab + b^2 = 2a^2 + 8ab + 10b^2
\]
We now add \( Kb^2 \):
\[
2a^2 + 8ab + (10 + K)b^2
\]
For this to be a perfect square, the middle term must be twice the product of the square root of the first and third terms:
\[
(2a + mb)^2 = 4a^2 + 4amb + m^2b^2
\]
Comparing terms, we see \( 8ab \) implies \( m = 4 \), and the last term gives \( m^2 = 10 + K \), so:
\[
16 = 10 + K \quad \Rightarrow \quad K = 6
\]

Thus, the value of \( K \) is \( 6 \).

---

Would you like further details on any part of this solution? Here are 5 related questions you could explore:

1. What are the common patterns in notable products used in these exercises?
2. How can cube roots be simplified when combined with irrational numbers?
3. How do you recognize perfect square trinomials?
4. What other types of products can be solved using notable product rules?
5. How does factoring play a role in simplifying complex expressions?

**Tip:** Always check for symmetries or patterns in algebraic expressions to simplify them faster.

Aplicando productos notables, calcular el valor de 16 × 3 × 5 × 17 × 257 + 1, y luego simplificar la expresión (³√(3 + 2√2 + ³√(3 - 2√2))³ - 9(³√(3 + 2√2 + ³√(3 - 2√2))).

Learn how to calculate the product of multiple numbers and simplify cube root expressions using notable products. This guide also covers how to find the value of K in a perfect square trinomial. Ideal for students in Grades 9-12.

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