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The problem asks us to find the value of the expression \( 2x - y + 3z - 2w \) based on the condition that the terms \( \frac{4}{9} m^4 n^7 q^3 r^3 \) and \( -8 m^x n^y q^z r^w \) are like terms ("semejantes" in Spanish).

To solve this, we need to recall that two terms are "like terms" when they have the same variables raised to the same powers.

### Step 1: Compare the terms
The terms are:
1. \( \frac{4}{9} m^4 n^7 q^3 r^3 \)
2. \( -8 m^x n^y q^z r^w \)

For these terms to be like terms, the exponents of corresponding variables \( m, n, q, r \) must be equal.

Thus, we equate the exponents:
- For \( m \): \( 4 = x \)
- For \( n \): \( 7 = y \)
- For \( q \): \( 3 = z \)
- For \( r \): \( 3 = w \)

### Step 2: Substitute values into the expression
Now that we know the values:
- \( x = 4 \)
- \( y = 7 \)
- \( z = 3 \)
- \( w = 3 \)

We substitute these into the expression \( 2x - y + 3z - 2w \):
\[
2(4) - 7 + 3(3) - 2(3)
\]
\[
= 8 - 7 + 9 - 6
\]
\[
= 4
\]

Thus, the value of the expression is \( \boxed{4} \).

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Would you like more details or have any questions?

Here are 5 related questions you might want to explore:
1. What is the general rule for identifying like terms in algebra?
2. How do coefficients affect the comparison of like terms?
3. Can we add or subtract unlike terms, and why not?
4. How does factoring help in simplifying algebraic expressions?
5. What are some common mistakes made when comparing algebraic terms?

**Tip:** When identifying like terms, always check both the variables and their exponents, but ignore the coefficients.

Si los términos: (4/9)m^4n^7q^3r^3 y -8m^x n^y q^z r^w son semejantes. Halle el valor de: 2x - y + 3z - 2w.

Learn how to solve an algebraic problem involving like terms and exponents. Given two terms (4/9)m^4n^7q^3r^3 and -8m^x n^y q^z r^w, find the value of the expression 2x - y + 3z - 2w. Ideal for students in Grades 8-10.

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