Let's go through each expression and simplify them:

1. $b^5 \cdot b^3$:

   • Using the rule for multiplying powers with the same base, $a^m \cdot a^n = a^{m+n}$,
   • $b^5 \cdot b^3 = b^{5+3} = b^8$.

   Answer: $b^8$.

2. $(-2)^4$:

   • Using the power rule, we calculate $(-2)^4$.
   • $(-2)^4 = (-2) \cdot (-2) \cdot (-2) \cdot (-2) = 16$.

   Answer: $16$.

3. $(m^3)^5$:

   • Using the power of a power rule, $(a^m)^n = a^{m \cdot n}$,
   • $(m^3)^5 = m^{3 \cdot 5} = m^{15}$.

   Answer: $m^{15}$.

4. $\frac{10m^{10}}{2m^6}$:

   • Simplify the coefficients: $\frac{10}{2} = 5$.
   • Using the rule for dividing powers with the same base, $\frac{a^m}{a^n} = a^{m-n}$,
   • $\frac{10m^{10}}{2m^6} = 5m^{10 - 6} = 5m^4$.

   Answer: $5m^4$.

5. $\left( \frac{r^2}{s^4} \right)^3$:

   • Apply the power of a quotient rule, $\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n}$,
   • $\left( \frac{r^2}{s^4} \right)^3 = \frac{(r^2)^3}{(s^4)^3} = \frac{r^{2 \cdot 3}}{s^{4 \cdot 3}} = \frac{r^6}{s^{12}}$.

   Answer: $\frac{r^6}{s^{12}}$.

Summary of Answers:

1. $b^8$
2. $16$
3. $m^{15}$
4. $5m^4$
5. $\frac{r^6}{s^{12}}$

Would you like further explanations for any specific step or have additional questions?

Follow-up Questions

1. What would be the result if the exponent in problem 1 was different, like $b^5 \cdot b^4$?
2. How would the answer to problem 2 change if the base was a different negative number, such as $(-3)^4$?
3. In problem 3, how does the result change if the base exponent was $m^4$ instead of $m^3$?
4. For problem 4, how would the result vary if the powers of $m$ were swapped, like $\frac{10m^6}{2m^{10}}$?
5. In problem 5, what happens if we change the fraction to $\left( \frac{r^3}{s^2} \right)^4$?

Tip

When simplifying expressions, always remember the rules of exponents: product of powers, power of a power, and power of a quotient. These rules help simplify calculations effectively.