https://r2.erweima.ai/mathbot/image/1726201176945cueujd0u.jpg

The image contains six expressions that seem to require simplification or factoring. I'll break them down and simplify each one step by step.

### 1. \( \frac{15a^2}{12ab} \)

Simplify by canceling out common factors:

- Cancel out \(a\) from numerator and denominator.
- \( \frac{15a^2}{12ab} = \frac{15a}{12b} \).
- \( \frac{15}{12} = \frac{5}{4} \).

The simplified form is:
\[
\frac{5a}{4b}
\]

### 2. \( \frac{14m^3n^3p^2}{-7m^2n^3p^2} \)

Simplify by canceling out common factors:

- \( \frac{14}{-7} = -2 \).
- \( m^3 \) and \( m^2 \) cancel to leave \( m \).
- \( n^3 \) cancels out completely.
- \( p^2 \) cancels out completely.

The simplified form is:
\[
-2m
\]

### 3. \( \frac{17p^3q^4z^6}{34p^7q^8z^{10}} \)

Simplify by canceling out common factors:

- \( \frac{17}{34} = \frac{1}{2} \).
- \( p^3 \) and \( p^7 \) cancel to leave \( \frac{1}{p^4} \).
- \( q^4 \) and \( q^8 \) cancel to leave \( \frac{1}{q^4} \).
- \( z^6 \) and \( z^{10} \) cancel to leave \( \frac{1}{z^4} \).

The simplified form is:
\[
\frac{1}{2p^4q^4z^4}
\]

### 4. \( \frac{21ab^3c^6}{35a^4b^2c^2} \)

Simplify by canceling out common factors:

- \( \frac{21}{35} = \frac{3}{5} \).
- \( a \) and \( a^4 \) cancel to leave \( \frac{1}{a^3} \).
- \( b^3 \) and \( b^2 \) cancel to leave \( b \).
- \( c^6 \) and \( c^2 \) cancel to leave \( c^4 \).

The simplified form is:
\[
\frac{3bc^4}{5a^3}
\]

### 5. \( \frac{p^3q - pq^3}{p^2q - pq^2} \)

Factor each part:

- Factor the numerator: \( p(p^2q - q^3) \).
- Factor the denominator: \( p(pq - q^2) \).

Now cancel the common factor \( p \) from the numerator and denominator:
\[
\frac{p^2q - q^3}{pq - q^2}
\]

This is the simplified form.

### 6. \( \frac{q^3 + 1}{q^4 + q - q^3 - 1} \)

First, factor both the numerator and denominator:

- The numerator is a sum of cubes: \( q^3 + 1 = (q + 1)(q^2 - q + 1) \).
- The denominator can be rearranged: \( q^4 - q^3 + q - 1 \). Factor this as \( (q + 1)(q^3 - 1) \).

The denominator \( q^3 - 1 \) is a difference of cubes: \( q^3 - 1 = (q - 1)(q^2 + q + 1) \).

Now simplify the fraction:
\[
\frac{(q + 1)(q^2 - q + 1)}{(q + 1)(q - 1)(q^2 + q + 1)}
\]
Cancel the \( (q + 1) \) terms:
\[
\frac{q^2 - q + 1}{(q - 1)(q^2 + q + 1)}
\]
This is the simplified form.

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Let me know if you need any further details or explanations! Here are some related questions:

1. How would you simplify \( \frac{x^4y^3}{x^2y^5} \)?
2. Can you factor \( x^2 - 25 \)?
3. What is the general method to simplify fractions with variables?
4. What is a sum of cubes and how do you factor it?
5. How do you deal with complex fractions with polynomial expressions?

**Tip:** When simplifying fractions with variables, always look for common factors that can be canceled out before performing any operations.

Simplify the following expressions: (1) 15a^2 / 12ab (2) 14m^3n^3p^2 / -7m^2n^3p^2 (3) 17p^3q^4z^6 / 34p^7q^8z^10 (4) 21ab^3c^6 / 35a^4b^2c^2 (5) (p^3q - pq^3) / (p^2q - pq^2) (6) (q^3 + 1) / (q^4 + q - q^3 - 1).

This problem set involves simplifying algebraic fractions and factorizing polynomials. It covers key algebraic concepts such as exponent rules, the sum and difference of cubes, and simplifying complex rational expressions. Suitable for students in Grades 9-11.

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