Make 20 exercises with properties of exponents.

Here are 20 exercises covering various properties of exponents, ranging from basic to more advanced problems:

### 1. **Simplify the expression:**
   \[
   2^3 \times 2^5
   \]

### 2. **Simplify the expression:**
   \[
   \frac{5^7}{5^3}
   \]

### 3. **Simplify the expression:**
   \[
   (3^4)^2
   \]

### 4. **Simplify the expression:**
   \[
   \frac{2^5 \times 2^3}{2^4}
   \]

### 5. **Evaluate the expression:**
   \[
   (-2)^4
   \]

### 6. **Simplify the expression:**
   \[
   x^3 \times x^6
   \]

### 7. **Simplify the expression:**
   \[
   (5x^2y^3)^2
   \]

### 8. **Simplify the expression:**
   \[
   \frac{x^7}{x^2}
   \]

### 9. **Simplify the expression:**
   \[
   (3^5 \times 2^3)^0
   \]

### 10. **Simplify the expression:**
   \[
   \left(\frac{a^4}{b^3}\right)^2
   \]

### 11. **Simplify the expression:**
   \[
   (4x^{-2})^3
   \]

### 12. **Simplify the expression:**
   \[
   2^3 \times 3^2 \times 2^{-1}
   \]

### 13. **Simplify the expression:**
   \[
   (x^{-4} y^5)^{-2}
   \]

### 14. **Simplify the expression:**
   \[
   \left( \frac{x^5 y^{-3}}{z^2} \right)^3
   \]

### 15. **Simplify the expression:**
   \[
   \frac{x^3 y^{-2}}{x^{-1} y^4}
   \]

### 16. **Simplify the expression:**
   \[
   (2^4 \times 3^2)^{-1}
   \]

### 17. **Simplify the expression:**
   \[
   \frac{2x^4}{4x^{-2}}
   \]

### 18. **Simplify the expression:**
   \[
   (2x^3 y^2 z^{-1})^2
   \]

### 19. **Evaluate the expression:**
   \[
   \left( \frac{1}{2} \right)^{-3}
   \]

### 20. **Simplify the expression:**
   \[
   \frac{x^{7/2}}{x^{3/2}}
   \]

These exercises cover various exponent properties such as the product rule, quotient rule, power of a power rule, zero exponents, negative exponents, and fractional exponents.

### Related Questions
1. What is the rule for multiplying exponents with the same base?
2. How does the negative exponent rule work?
3. What happens when an exponent is zero?
4. How do you simplify expressions with fractional exponents?
5. How does the power of a quotient rule apply in exponents?

### Tip
Always remember to combine like bases by adding or subtracting exponents, and apply the rules for powers carefully when dealing with multiple terms!

Explore 20 exercises covering the properties of exponents, including rules for multiplying, dividing, and raising powers, as well as negative and zero exponents. These problems are suitable for students in Grades 8-10 and include both basic and advanced exponent concepts.

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