https://r2.erweima.ai/mathbot/image/1726266410335r027wlmi.jpg

Let's analyze the expressions given in the image.

We have two expressions:

1. \( M(m; n) = \frac{3}{5} m^{4+y} n^8 \)
2. \( N(m; n) = 3a^2 m^7 n^{x+5} \)

The problem asks to find the value of \( y + x \), assuming these terms are "términos semejantes" (like terms). Like terms in algebra must have the same variables raised to the same powers.

### Step-by-step process:

1. Compare the powers of \( m \) in both expressions:
   - In \( M(m; n) \), the power of \( m \) is \( 4 + y \).
   - In \( N(m; n) \), the power of \( m \) is 7.
   
   Since the terms must be like terms, the exponents must be equal. Therefore:
   \[
   4 + y = 7
   \]
   Solving for \( y \):
   \[
   y = 7 - 4 = 3
   \]

2. Compare the powers of \( n \) in both expressions:
   - In \( M(m; n) \), the power of \( n \) is 8.
   - In \( N(m; n) \), the power of \( n \) is \( x + 5 \).
   
   Again, since the terms must be like terms, the exponents must be equal. Therefore:
   \[
   8 = x + 5
   \]
   Solving for \( x \):
   \[
   x = 8 - 5 = 3
   \]

### Final Answer:
Since \( y = 3 \) and \( x = 3 \), we have:
\[
y + x = 3 + 3 = 6
\]

Thus, the value of \( y + x \) is 6.

---

Would you like more details or have any other questions?

Here are five related questions to deepen your understanding:

1. What defines "like terms" in an algebraic expression?
2. How can exponents be compared when dealing with variables?
3. Why is it necessary for like terms to have identical powers of variables?
4. How would the problem change if the coefficients (like \( \frac{3}{5} \) and \( 3a^2 \)) also needed to match?
5. What happens if the exponents of one variable match, but the other doesn't?

**Tip**: Always verify that all corresponding variable terms (both bases and exponents) match when determining if terms are alike in algebraic expressions.

Si: M(m;n) = (3/5)m^(4+y)n^8 y N(m;n) = 3a^2m^7n^(x+5), son términos semejantes. Halle: y + x.

Learn how to solve for y and x in the algebraic expressions M(m;n) = (3/5)m^(4+y)n^8 and N(m;n) = 3a^2m^7n^(x+5) by equating the exponents for like terms. Detailed explanation for students in Grades 10-12.

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