Exponent Rules: Power of a Product
Overview
The Power of a Product rule states that when raising a product to a power, each factor in the product is raised to the same power. In mathematical terms:
\((ab)^n = a^n \cdot b^n\)
This rule applies when a product inside parentheses is raised to an exponent.
Steps to Apply the Power of a Product Rule
- Distribute the exponent to each factor in the product.
- Simplify the resulting expressions, if possible.
Example 1: Simplifying with Numbers
Consider the following expression:
\((2 \cdot 3)^4\)
Step 1: Distribute the exponent 4 to each factor:
\(2^4 \cdot 3^4\)
Step 2: Simplify each term:
\(16 \cdot 81 = 1296\)
Thus, the simplified expression is:
\(1296\)
Example 2: Simplifying with Variables
Consider the following expression:
\((x \cdot y)^3\)
Step 1: Distribute the exponent 3 to each variable:
\(x^3 \cdot y^3\)
Thus, the simplified expression is:
\(x^3 \cdot y^3\)
Practice Questions
- Question 1: Simplify the following expression:
\((2 \cdot 5)^3\)
Solution
Step 1: Distribute the exponent 3 to each factor:
\(2^3 \cdot 5^3\)
Step 2: Simplify:
\(8 \cdot 125 = 1000\)
- Question 2: Simplify the following expression:
\((a \cdot b)^5\)
Solution
Step 1: Distribute the exponent 5 to each variable:
\(a^5 \cdot b^5\)
- Question 3: Simplify the following expression:
\((3 \cdot x)^2\)
Solution
Step 1: Distribute the exponent 2 to each factor:
\(3^2 \cdot x^2\)
Step 2: Simplify:
\(9 \cdot x^2 = 9x^2\)
- Question 4: Simplify the following expression:
\((m \cdot n)^4\)
Solution
Step 1: Distribute the exponent 4 to each variable:
\(m^4 \cdot n^4\)
- Question 5: Simplify the following expression:
\((4 \cdot y)^3\)
Solution
Step 1: Distribute the exponent 3 to each factor:
\(4^3 \cdot y^3\)
Step 2: Simplify:
\(64 \cdot y^3 = 64y^3\)