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The problem asks to find the greatest common divisor (GCD) of the two algebraic expressions:

\[
12w^6 u^8 y^2 \quad \text{and} \quad 30u^7 y^3
\]

### Step-by-step solution:

#### 1. Coefficients (Numerical Part):
We first find the GCD of the numerical coefficients, which are 12 and 30.

- The prime factorization of 12 is \(2^2 \times 3\).
- The prime factorization of 30 is \(2 \times 3 \times 5\).

The common factors between 12 and 30 are \(2\) and \(3\), so the GCD of 12 and 30 is \(2 \times 3 = 6\).

#### 2. Variables:
Now, we find the GCD for each variable separately.

- For \(w\):
  - In the first expression, \(w^6\) appears.
  - In the second expression, there is no \(w\) term.
  
  Since there is no \(w\) term in the second expression, the GCD for \(w\) is \(w^0 = 1\).

- For \(u\):
  - In the first expression, \(u^8\) appears.
  - In the second expression, \(u^7\) appears.
  
  The GCD is the smallest exponent, which is \(u^7\).

- For \(y\):
  - In the first expression, \(y^2\) appears.
  - In the second expression, \(y^3\) appears.
  
  The GCD is the smallest exponent, which is \(y^2\).

#### 3. Final GCD Expression:
Now, we combine everything together:

\[
\text{GCD} = 6u^7 y^2
\]

This is the greatest common divisor of the two given algebraic expressions.

Do you need further details or explanations?

Here are 5 related questions you could explore:
1. How is the GCD of two numbers calculated using their prime factorizations?
2. Can the GCD of two algebraic expressions include terms like \(w\) if one of the expressions lacks it?
3. How would you find the least common multiple (LCM) of these two expressions?
4. What happens to the GCD if we change the exponents of \(y\) in one expression?
5. How would the GCD change if the second expression had a higher exponent for \(u\) than the first?

**Tip:** When finding the GCD of algebraic expressions, always take the smallest exponent for each variable common to both terms!

Hallar el máximo común divisor de las expresiones 12w^6 u^8 y^2 y 30u^7 y^3.

Learn how to find the greatest common divisor (GCD) of two algebraic expressions: 12w^6 u^8 y^2 and 30u^7 y^3. This step-by-step solution involves prime factorization and choosing the smallest exponent for each variable. Suitable for high school students in Grades 9-12.

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