3. As the exponent decreases, what happens to the value of the expression?

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UpStudy AI · Accepted Answer

The effect of decreasing the exponent on the value of an expression depends primarily on the **base** of the exponentiation. Here's a detailed breakdown:

### 1. **Base Greater Than 1 (e.g., 2, 3, 10)**
When the base is **greater than 1**, the expression **increases** as the exponent **increases** and **decreases** as the exponent **decreases**.

**Example:**
- **Base = 2**
  - $$2^5 = 32$$
  - $$2^3 = 8$$ (Exponent decreased from 5 to 3; Value decreased)
  - $$2^1 = 2$$ (Exponent decreased from 3 to 1; Value decreased)

### 2. **Base Between 0 and 1 (e.g., 1/2, 1/3)**
When the base is **between 0 and 1**, the expression **decreases** as the exponent **increases** and **increases** as the exponent **decreases**.

**Example:**
- **Base = 1/2**
  - $$(1/2)^5 = 1/32$$
  - $$(1/2)^3 = 1/8$$ (Exponent decreased from 5 to 3; Value increased)
  - $$(1/2)^1 = 1/2$$ (Exponent decreased from 3 to 1; Value increased)

### 3. **Base Equal to 1**
If the base is **exactly 1**, the value of the expression **remains the same** regardless of the exponent.

**Example:**
- **Base = 1**
  - $$1^5 = 1$$
  - $$1^3 = 1$$
  - $$1^1 = 1$$ (Value remains constant)

### 4. **Negative Bases**
For **negative bases**, the situation is more complex because the sign of the result depends on whether the exponent is even or odd. However, the general trend regarding magnitude (ignoring sign) follows the rules above based on the absolute value of the base.

**Example:**
- **Base = -2**
  - $$(-2)^5 = -32$$
  - $$(-2)^3 = -8$$ (Exponent decreased from 5 to 3; Magnitude decreased)
  - $$(-2)^1 = -2$$ (Exponent decreased from 3 to 1; Magnitude decreased)

### **Summary**
- **Base > 1:**  
  *Decreasing the exponent ⇒ Decreasing the expression's value.*
  
- **0 < Base < 1:**  
  *Decreasing the exponent ⇒ Increasing the expression's value.*
  
- **Base = 1:**  
  *Expression's value remains unchanged regardless of the exponent.*

- **Negative Bases:**  
  *Magnitude behaves like positive bases based on the absolute value, but the sign depends on whether the exponent is even or odd.*

Understanding the relationship between the base and the exponent is crucial for predicting how changes in the exponent will affect the overall value of the expression.
When the exponent decreases, the value of the expression decreases if the base is greater than 1, and increases if the base is between 0 and 1. If the base is 1, the value stays the same.

3. As the exponent decreases, what happens to the value of the expression?

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