https://file.aiquickdraw.com/mathbot/file/17309180606823ug66u49.jpg

I’ll go through each expression and provide detailed solutions.

### 1. \( b^5 \cdot b^3 \)
Using the product of powers rule, \( a^m \cdot a^n = a^{m+n} \), we get:
\[
b^5 \cdot b^3 = b^{5+3} = b^8
\]

### 2. \( \left( \frac{2}{5^4} \right)^3 \)
To simplify this, apply the power of a quotient rule, \( \left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \):
\[
\left( \frac{2}{5^4} \right)^3 = \frac{2^3}{(5^4)^3}
\]
Calculating the powers:
\[
2^3 = 8 \quad \text{and} \quad (5^4)^3 = 5^{4 \cdot 3} = 5^{12}
\]
So, the simplified form is:
\[
\frac{8}{5^{12}}
\]

### 3. \( (-2)^3 \)
To find this, simply calculate the power:
\[
(-2)^3 = -2 \cdot -2 \cdot -2 = -8
\]

### 4. \( \frac{10m^6}{2m^{10}} \)
Divide the coefficients and apply the quotient of powers rule, \( \frac{a^m}{a^n} = a^{m-n} \):
\[
\frac{10}{2} = 5 \quad \text{and} \quad \frac{m^6}{m^{10}} = m^{6-10} = m^{-4}
\]
The answer is:
\[
5m^{-4} = \frac{5}{m^4} \quad \text{(writing with a positive exponent)}
\]

### 5. \( (m^3)^5 \)
Apply the power of a power rule, \( (a^m)^n = a^{m \cdot n} \):
\[
(m^3)^5 = m^{3 \cdot 5} = m^{15}
\]

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### Final Answers:
1. \( b^8 \)
2. \( \frac{8}{5^{12}} \)
3. \( -8 \)
4. \( \frac{5}{m^4} \)
5. \( m^{15} \)

Would you like further explanations on any of these steps, or do you have any additional questions?

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#### Related Questions:
1. How do you apply the power of a quotient rule in other contexts?
2. What is the product of powers rule, and when is it used?
3. Why is it necessary to express answers with positive exponents in some contexts?
4. How do negative exponents simplify expressions in division problems?
5. How does the power of a

Explore step-by-step solutions for simplifying expressions involving zero, negative, and rational exponents. This guide uses key exponent rules, such as the product of powers and power of a quotient, to solve common algebraic expressions for students in Grades 8-10.

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