https://r2.erweima.ai/mathbot/image/1726346981761ww5tmwjh.jpg

Let's address the exercises one by one from the image:

### 3. Determinar qué parejas de términos son semejantes. Explicar la respuesta.

To determine if two terms are *semejantes* (similar), they must have the same variables raised to the same powers. The coefficients do not need to be the same for terms to be similar.

Let's analyze each pair:

- **a. \( \sigma \), \( 3\sigma \)**  
  These terms are similar because they both contain the same variable \( \sigma \) (with an implicit exponent of 1). Only the coefficients differ.

- **b. \( 5xy \), \( -2xyz \)**  
  These terms are not similar because the second term has an additional variable \( z \), making their variable parts different.

- **c. \( -11mn \), \( -2mn \)**  
  These terms are similar because both have the variables \( m \) and \( n \) raised to the same powers (1 each).

- **d. \( -5x^2y \), \( 2yx^2 \)**  
  These terms are similar because both have the variables \( x^2y \), though the order of variables differs, which does not affect their similarity.

- **e. \( 8x^2y \), \( -3yx^2 \)**  
  These terms are also similar because both have \( x^2y \), with different coefficients.

- **f. \( 2x^2y \), \( -5xy^2z^2 \)**  
  These terms are not similar because the second term has an extra \( z^2 \) and the powers of \( y \) are different.

- **g. \( \frac{5}{6}a^2b \), \( -a^2b \)**  
  These terms are similar because they have the same variables \( a^2b \) and differ only by their coefficients.

- **h. \( -14x^6y \), \( 3x^4y \)**  
  These terms are not similar because the exponents of \( x \) are different (\( x^6 \) vs. \( x^4 \)).

- **i. \( -85x^a y^b \), \( y^a x^b \)**  
  These terms are similar because both have \( x^a y^b \), though the order is different.

- **j. \( 5m^2n^3a^2 \), \( -2m^3an^2p \)**  
  These terms are not similar because they have different exponents for \( m \), \( n \), and different variables (one has \( p \)).

### 4. Escribir dos monomios que sean semejantes a cada uno de los monomios dados.

To form similar monomials, we must keep the same variables with the same exponents.

- **a. \( 4\alpha \)**  
  Examples of similar monomials:
  - \( 7\alpha \)
  - \( -2\alpha \)

- **b. \( -5x^3 \)**  
  Examples of similar monomials:
  - \( 3x^3 \)
  - \( -x^3 \)

- **c. \( 0.7x^2y \)**  
  Examples of similar monomials:
  - \( 5x^2y \)
  - \( -2x^2y \)

- **d. \( 4n^4m \)**  
  Examples of similar monomials:
  - \( -7n^4m \)
  - \( 2n^4m \)

- **e. \( \frac{1}{9}pq^2 \)**  
  Examples of similar monomials:
  - \( 3pq^2 \)
  - \( -\frac{5}{2}pq^2 \)

- **f. \( \frac{-3}{5}xyz^3 \)**  
  Examples of similar monomials:
  - \( 2xyz^3 \)
  - \( -\frac{7}{5}xyz^3 \)

- **g. \( \sqrt{3}bc \)**  
  Examples of similar monomials:
  - \( 4bc \)
  - \( \frac{-\sqrt{5}}{2}bc \)

- **h. \( 2abx \)**  
  Examples of similar monomials:
  - \( -abx \)
  - \( 3abx \)

- **i. \( -13pqr^3 \)**  
  Examples of similar monomials:
  - \( 5pqr^3 \)
  - \( -7pqr^3 \)

- **j. \( 2n \)**  
  Examples of similar monomials:
  - \( -4n \)
  - \( 6n \)

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Let me know if you'd like more details about any of these steps!

### Related Questions:
1. What are the key characteristics that define two similar terms?
2. How do coefficients affect the similarity of terms?
3. Can two terms be considered similar if they differ in the number of variables? Why or why not?
4. How would you combine like terms in a polynomial?
5. What happens when terms are ordered differently? Does it affect their similarity?

### Tip:
When checking for similarity of terms, always focus on the variable part (both the variables and their exponents) and ignore the coefficients.

Determinar qué parejas de términos son semejantes y escribir dos monomios semejantes a cada uno de los monomios dados.

Aprende a identificar términos semejantes en expresiones algebraicas y a escribir monomios semejantes. Este problema cubre conceptos clave de álgebra, incluyendo cómo determinar la semejanza de términos según sus variables y exponentes.

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