https://file.aiquickdraw.com/mathbot/file/1730772888965eva8qn8j.jpg

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:** \( \frac{r^6}{s^{12}} \).

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### 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?

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### 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 \( \left( \frac{r^3}{s^2} \right)^4 \)?

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### 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.

Simplify the following expressions:
1. \( b^5 \cdot b^3 \)
2. \( (-2)^4 \)
3. \( (m^3)^5 \)
4. \( \frac{10m^{10}}{2m^6} \)
5. \( \left( \frac{r^2}{s^4} \right)^3 \)

Learn how to simplify expressions involving exponents using multiplication, division, and power rules. This guide covers problems like \( b^5 \cdot b^3 \), \( (-2)^4 \), \( (m^3)^5 \), and more, ideal for students in Grades 9-10.

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