AP Precalculus Practice Test: Unit 2 Question #6 Product of Numbers into an Exponential Function

My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May.

### AP Precalculus Practice Test: Unit 2, Question #6
**Topic:** Product of Numbers Represented as an Exponential Function

This question involves expressing a product of numbers in the form of an exponential function by recognizing patterns and simplifying expressions.

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### **Example Problem:**
Write the product \( 3 \cdot 3^2 \cdot 3^4 \) as a single exponential expression.

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### **Solution:**

1. **Recall the properties of exponents:**
- When multiplying terms with the same base, add their exponents:
\[
a^m \cdot a^n = a^{m+n}
\]

2. **Identify the base and exponents:**
- The base is \(3\).
- The exponents are \(1\), \(2\), and \(4\) (note that \(3\) can be written as \(3^1\)).

3. **Add the exponents:**
\[
3^1 \cdot 3^2 \cdot 3^4 = 3^{1+2+4}
\]
\[
3^1 \cdot 3^2 \cdot 3^4 = 3^7
\]

4. **Final Answer:**
\[
3^7
\]

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### **General Approach:**

1. **Identify the base:**
All terms in the product must have the same base.

2. **Add the exponents:**
Use the property of exponents to simplify the product into a single exponential term.

3. **Simplify the expression:**
Combine terms and write the result in the form \(a^n\).

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### **Example Problem 2:**
Express \( 2^3 \cdot 2 \cdot 2^5 \cdot 2^2 \) as a single exponential expression.

**Solution:**
1. Rewrite \(2\) as \(2^1\):
\[
2^3 \cdot 2^1 \cdot 2^5 \cdot 2^2
\]

2. Add the exponents:
\[
2^{3+1+5+2} = 2^{11}
\]

**Final Answer:**
\[
2^{11}
\]

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### **Potential Multiple-Choice Options (Example 1):**
A. \(3^6\)
B. \(3^7\)
C. \(3^9\)
D. \(3^8\)

(Correct Answer: **B**)

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### **Applications of the Concept:**
1. **Simplifying Data Models:**
- Combining repeated growth factors in exponential models.
- Example: Population growth or interest rates.

2. **Exponential Function Simplification:**
- Preparing expressions for graphing or solving equations.

This question reinforces understanding of exponents, a foundational concept for exponential functions and their applications.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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