Here’s a step-by-step breakdown for problems 10 to 18:

10. $a^3 \div a^2$

Solution:
When dividing like bases, subtract the exponents:
$a^3 \div a^2 = a^{3-2} = a^1 = a.$

11. $2^{-3}$

Solution:
A negative exponent means the reciprocal of the base raised to the positive exponent:
$2^{-3} = \frac{1}{2^3} = \frac{1}{8}.$

12. $3^2 \cdot 3^4$

Solution:
When multiplying like bases, add the exponents:
$3^2 \cdot 3^4 = 3^{2+4} = 3^6.$

13. $3 \cdot x^3 \cdot x^3$

Solution:
Combine the exponents of like bases:
$3 \cdot x^3 \cdot x^3 = 3 \cdot x^{3+3} = 3x^6.$

14. $2^{m+3} \div 2^m$

Solution:
When dividing like bases, subtract the exponents:
$2^{m+3} \div 2^m = 2^{(m+3) - m} = 2^3 = 8.$

15. $2^3 \cdot 2^4$

Solution:
When multiplying like bases, add the exponents:
$2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128.$

16. $(2a)^3 \cdot (3a^2)^2$

Solution:
First expand the powers for each term:
$(2a)^3 = 2^3 \cdot a^3 = 8a^3, \quad (3a^2)^2 = 3^2 \cdot a^{2 \cdot 2} = 9a^4.$
Now multiply:
$8a^3 \cdot 9a^4 = (8 \cdot 9) \cdot a^{3+4} = 72a^7.$

17. $(7m^2)^3$

Solution:
Raise each factor inside the parentheses to the power of 3:
$(7m^2)^3 = 7^3 \cdot (m^2)^3 = 343 \cdot m^{2 \cdot 3} = 343m^6.$

18. $2x^4 \cdot (3x^2)^2 \div (x^2 \cdot 2)^2$

Solution:

1. Expand each term:
   $(3x^2)^2 = 3^2 \cdot x^{2 \cdot 2} = 9x^4, \quad (x^2 \cdot 2)^2 = x^{2 \cdot 2} \cdot 2^2 = x^4 \cdot 4.$
2. Multiply and divide step by step:
   $2x^4 \cdot 9x^4 = 18x^{4+4} = 18x^8.$
   Divide by $x^4 \cdot 4$:
   $\frac{18x^8}{4x^4} = \frac{18}{4} \cdot x^{8-4} = \frac{9}{2}x^4.$

Final Answers:

10. $a$
11. $\frac{1}{8}$
12. $3^6$
13. $3x^6$
14. $8$
15. $128$
16. $72a^7$
17. $343m^6$
18. $\frac{9}{2}x^4$

Let me know if you need further clarification on any step!
Related Questions:

1. How do you simplify powers of powers?
2. What are the rules for negative exponents?
3. How do you combine terms with like bases?
4. What is the general formula for raising a product to a power?
5. How do you divide powers with the same base?

Tip: Always expand expressions step by step when dealing with powers and parentheses!