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Let's solve each of these problems one by one.

### 1. First Problem: 

We are asked to find the **greatest common divisor (GCD)** of the expressions:

\[
18x^5, \ 30x^2, \ 12x
\]

#### Step 1: Find the GCD of the coefficients (18, 30, and 12).
- Prime factorization of 18: \( 18 = 2 \times 3^2 \)
- Prime factorization of 30: \( 30 = 2 \times 3 \times 5 \)
- Prime factorization of 12: \( 12 = 2^2 \times 3 \)

The GCD of the coefficients is:
\[
\text{GCD}(18, 30, 12) = 2 \times 3 = 6
\]

#### Step 2: Find the GCD of the powers of \(x\).
- The powers of \(x\) are 5, 2, and 1. The GCD of these powers is the smallest power, which is 1.

So, the greatest common divisor of the three expressions is:
\[
6x
\]

### 2. Second Problem: 

We are asked to find the **GCD** of the expressions:

\[
12y^7x^5v^3 \ \text{and} \ 20y^8x^5
\]

#### Step 1: Find the GCD of the coefficients (12 and 20).
- Prime factorization of 12: \( 12 = 2^2 \times 3 \)
- Prime factorization of 20: \( 20 = 2^2 \times 5 \)

The GCD of the coefficients is:
\[
\text{GCD}(12, 20) = 2^2 = 4
\]

#### Step 2: Find the GCD of the powers of each variable:
- For \(y^7\) and \(y^8\), the GCD is \(y^7\) (the lower power).
- For \(x^5\) and \(x^5\), the GCD is \(x^5\).
- For \(v^3\) and the absence of \(v\) in the second expression, the GCD is \(v^0 = 1\).

So, the GCD of the two expressions is:
\[
4y^7x^5
\]

---

Do you have any questions about these solutions or would you like further details?

Here are 5 related questions:
1. How do you find the least common multiple (LCM) of algebraic expressions?
2. Can the greatest common divisor be greater than the smallest coefficient?
3. How does factoring help in finding the GCD of expressions?
4. How would you find the GCD of more complex polynomials?
5. What is the GCD if none of the terms share common variables?

**Tip:** When finding the GCD of algebraic expressions, always factor both the coefficients and the variables separately.

Hallar el máximo común divisor de las expresiones 18x^5, 30x^2, y 12x y Hallar el máximo común divisor de las expresiones 12y^7x^5v^3 y 20y^8x^5

Learn how to find the greatest common divisor (GCD) of algebraic expressions such as 18x^5, 30x^2, and 12x. This guide covers the step-by-step process of using prime factorization and variable powers. Additionally, explore the GCD calculation for 12y^7x^5v^3 and 20y^8x^5. Suitable for students in Grades 9-12.

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