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The question visible in the image pertains to simplifying and evaluating powers, particularly related to multiplying and dividing powers. Let's go through each of the problems:

1. **a. \( \frac{x^2 \cdot x^5}{x^1} \)**  
   Using the law of exponents: \( a^m \cdot a^n = a^{m+n} \) and \( \frac{a^m}{a^n} = a^{m-n} \):
   - First, simplify the numerator: \( x^2 \cdot x^5 = x^{2+5} = x^7 \)
   - Now, divide by \( x^1 \):  
     \( \frac{x^7}{x^1} = x^{7-1} = x^6 \)

2. **b. \( \frac{y^{10}}{y^6} \)**  
   Using the division law for exponents \( \frac{a^m}{a^n} = a^{m-n} \):
   \[
   \frac{y^{10}}{y^6} = y^{10-6} = y^4
   \]

3. **c. \( (2^3)(2^3) \)**  
   Applying the multiplication rule: \( a^m \cdot a^n = a^{m+n} \):
   \[
   2^3 \cdot 2^3 = 2^{3+3} = 2^6
   \]
   Evaluate \( 2^6 \):  
   \( 2^6 = 64 \)

4. **d. \( \frac{8^5}{8^3} \)**  
   Using the division law for exponents:
   \[
   \frac{8^5}{8^3} = 8^{5-3} = 8^2
   \]
   Evaluate \( 8^2 \):  
   \( 8^2 = 64 \)

5. **e. \( a^3 \cdot a \)**  
   Here, we recognize that \( a \) is the same as \( a^1 \), so apply the multiplication rule:
   \[
   a^3 \cdot a = a^{3+1} = a^4
   \]

These are the simplified expressions and evaluations where possible.

Let me know if you'd like any further details on any of these or have questions about the steps!

### 5 related questions:
1. How do you handle powers with different bases when multiplying or dividing?
2. What happens when you raise a power to another power?
3. How would the rules change for negative exponents?
4. Can these rules be applied to fractional bases or exponents?
5. How can you simplify expressions involving multiple variables with exponents?

### Tip:
When multiplying or dividing powers, always focus on simplifying each step using the rules of exponents, which can make seemingly complex expressions much easier to handle.

Simplify and evaluate expressions involving powers, including multiplication and division of powers.

Learn how to simplify and evaluate expressions involving exponents using the laws of exponents. Problems include multiplying and dividing powers. Suitable for students in Grades 7-9.

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