AP Precalculus Practice Test: Unit 2 Question #8 Rewriting a Power Using Exponential Properties Ex 1

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 #8
**Topic:** Rewriting a Power Using Exponential Properties

This question tests your ability to simplify expressions with exponents using the rules of exponents.

---

### **Example Problem 1:**
Simplify the expression:
\[
(x^3)^4
\]

---

### **Solution:**

1. **Recall the power rule for exponents:**
\[
(a^m)^n = a^{m \cdot n}
\]
This rule states that when raising a power to another power, you multiply the exponents.

2. **Apply the rule to the given expression:**
\[
(x^3)^4 = x^{3 \cdot 4}
\]

3. **Simplify the exponent:**
\[
x^{3 \cdot 4} = x^{12}
\]

---

**Final Answer:**
\[
x^{12}
\]

---

### **Additional Example Problem:**
Simplify:
\[
\left(2^5\right)^3
\]

**Solution:**

1. Use the power rule:
\[
\left(2^5\right)^3 = 2^{5 \cdot 3}
\]

2. Simplify the exponent:
\[
2^{5 \cdot 3} = 2^{15}
\]

**Final Answer:**
\[
2^{15}
\]

---

### **Key Exponent Rules to Remember:**

1. **Product of Powers Rule:**
\[
a^m \cdot a^n = a^{m+n}
\]

2. **Quotient of Powers Rule:**
\[
\frac{a^m}{a^n} = a^{m-n}, \quad a \neq 0
\]

3. **Power of a Power Rule:**
\[
(a^m)^n = a^{m \cdot n}
\]

4. **Power of a Product Rule:**
\[
(ab)^m = a^m \cdot b^m
\]

5. **Power of a Quotient Rule:**
\[
\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}, \quad b \neq 0
\]

---

### Practice Problem for Review:

Simplify the following expressions:
1. \((x^2)^5\)
2. \((3y^4)^3\)
3. \(\left(\frac{a^2}{b^3}\right)^2\)

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Understanding and mastering these properties will simplify your work with exponents, a key skill in AP Precalculus and beyond!

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

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