Let's solve each of the problems step-by-step, using exponent rules.

A) $x^3 \cdot x^7 = x^p$

Rule: When multiplying terms with the same base, add their exponents.

$x^3 \cdot x^7 = x^{3 + 7} = x^{10}$

Thus, $p = 10$.

B) $\left( x^3 \right)^7 = x^r$

Rule: When raising a power to another power, multiply the exponents.

$\left( x^3 \right)^7 = x^{3 \cdot 7} = x^{21}$

Thus, $r = 21$.

C) $\frac{x^7}{x^3} = x^q$

Rule: When dividing terms with the same base, subtract the exponents.

$\frac{x^7}{x^3} = x^{7 - 3} = x^4$

Thus, $q = 4$.

Final Answers:

• $p = 10$
• $r = 21$
• $q = 4$

Would you like more details or explanations?
Here are some questions to expand on this:

1. What are other exponent rules for operations with exponents?
2. How do negative exponents affect multiplication and division rules?
3. How do fractional exponents behave with the same base operations?
4. Can you provide examples of these exponent rules with variables?
5. How would the rules change if you had different bases for multiplication or division?

Tip: Remember that any base raised to the power of 0 is always 1 (e.g., $x^0 = 1$).