Full Prep Academy

Overview

The Product of Powers rule states that when multiplying powers with the same base, you add the exponents. In mathematical terms:

\(\ a^m \cdot a^n = a^{m+n} \)

This rule applies only when the base is the same in both terms. The exponents are added together to simplify the expression.

Steps to Apply the Product of Powers Rule

1. Identify the common base in both terms.
2. Add the exponents together.
3. Write the base and the sum of the exponents as a single power.

Example 1: Simplifying with Same Base

Consider the following expression:

\( 3^4 \cdot 3^2 \)

Step 1: The base is \( 3 \), so we add the exponents 4 and 2:

\( 3^{4+2} = 3^6 \)

Thus, the simplified expression is:

\( 3^6 \)

Example 2: Simplifying with Variables

Consider the following expression:

\( x^5 \cdot x^3 \)

Step 1: The base is \( x \), so we add the exponents 5 and 3:

\( x^{5+3} = x^8 \)

Thus, the simplified expression is:

\( x^8 \)

Practice Questions

1. Question 1: Simplify the following expression:

   \( 2^7 \cdot 2^4 \)

   Solution

   Step 1: The base is \( 2 \), so we add the exponents 7 and 4:

   \( 2^{7+4} = 2^{11} \)

2. Question 2: Simplify the following expression:

   \( y^6 \cdot y^2 \)

   Solution

   Step 1: The base is \( y \), so we add the exponents 6 and 2:

   \( y^{6+2} = y^8 \)

3. Question 3: Simplify the following expression:

   \( 5^3 \cdot 5^5 \)

   Solution

   Step 1: The base is \( 5 \), so we add the exponents 3 and 5:

   \( 5^{3+5} = 5^8 \)

4. Question 4: Simplify the following expression:

   \( a^9 \cdot a^3 \)

   Solution

   Step 1: The base is \( a \), so we add the exponents 9 and 3:

   \( a^{9+3} = a^{12} \)

5. Question 5: Simplify the following expression:

   \( 7^2 \cdot 7^4 \)

   Solution

   Step 1: The base is \( 7 \), so we add the exponents 2 and 4:

   \( 7^{2+4} = 7^6 \)