Let's go through each problem one by one and simplify the given expressions.

b. $\frac{30p^{10} \times 5q^6}{20p^3}$

1. First, simplify the coefficients:
   $\frac{30 \times 5}{20} = \frac{150}{20} = 7.5$

2. For the $p$ terms:
   $\frac{p^{10}}{p^3} = p^{10-3} = p^7$

3. Combine everything:
   $7.5 p^7 q^6$

c. $\frac{10x^2 \times 6y^3}{5x \times 12y^4}$

1. Simplify the coefficients:
   $\frac{10 \times 6}{5 \times 12} = \frac{60}{60} = 1$

2. Simplify the $x$ terms:
   $\frac{x^2}{x} = x^{2-1} = x$

3. Simplify the $y$ terms:
   $\frac{y^3}{y^4} = y^{3-4} = y^{-1} = \frac{1}{y}$

4. Combine everything:
   $\frac{x}{y}$

d. $v \times \frac{7}{v^2}$

1. Simplify the $v$ terms:
   $v \times \frac{1}{v^2} = \frac{v}{v^2} = \frac{1}{v}$

2. Combine everything:
   $\frac{7}{v}$

e. $\frac{14x^4 y^3}{7x^5 \times 21y^2}$

1. Simplify the coefficients:
   $\frac{14}{7 \times 21} = \frac{14}{147} = \frac{2}{21}$

2. Simplify the $x$ terms:
   $\frac{x^4}{x^5} = x^{4-5} = x^{-1} = \frac{1}{x}$

3. Simplify the $y$ terms:
   $\frac{y^3}{y^2} = y^{3-2} = y$

4. Combine everything:
   $\frac{2y}{21x}$

Final Answers:

• b. $7.5 p^7 q^6$
• c. $\frac{x}{y}$
• d. $\frac{7}{v}$
• e. $\frac{2y}{21x}$

Would you like more details on any step?

Here are some related questions:

1. How do you simplify expressions with variables in both the numerator and denominator?
2. Can you explain the properties of exponents used in these examples?
3. How do you handle negative exponents?
4. What are the general steps for simplifying complex rational expressions?
5. How would these problems change if there were additional terms inside parentheses?

Tip: Always simplify coefficients first, then handle the variables using exponent rules.