Let's simplify each expression step-by-step, using the property of exponents $(a^m)^n = a^{m \cdot n}$ where you multiply the exponents.

1. $(x^4)^5$
   Using the exponent rule, we multiply the exponents:
   $(x^4)^5 = x^{4 \cdot 5} = x^{20}$

2. $(y^3)^2$
   Similarly, multiply the exponents:
   $(y^3)^2 = y^{3 \cdot 2} = y^6$

3. $(x^2)^7$
   Again, multiply the exponents:
   $(x^2)^7 = x^{2 \cdot 7} = x^{14}$

4. $(x^3)^6 \cdot x^{20}$
   First, simplify $(x^3)^6$ by multiplying the exponents:
   $(x^3)^6 = x^{3 \cdot 6} = x^{18}$ Now, use the property of exponents that $x^m \cdot x^n = x^{m+n}$:
   $x^{18} \cdot x^{20} = x^{18+20} = x^{38}$

Final answers:

1. $(x^4)^5 = x^{20}$
2. $(y^3)^2 = y^6$
3. $(x^2)^7 = x^{14}$
4. $(x^3)^6 \cdot x^{20} = x^{38}$

Would you like further details or examples on any of these?

Follow-up questions:

1. Can you explain how the rule $(a^m)^n = a^{m \cdot n}$ works with a more complex base?
2. How does this rule apply to negative exponents or fractional exponents?
3. What happens if you divide expressions with exponents, like $\frac{x^5}{x^2}$?
4. What would happen if you had a negative base, such as $(-x^4)^5$?
5. How does the exponent rule apply to variables raised to powers in algebraic equations?

Tip:

When multiplying exponents with the same base, remember to add the exponents. This is useful for simplifying expressions in algebra.