Lesson 8.3
The Power Rule
What happens when an exponent gets raised to ANOTHER exponent? The power levels multiply.
Introduction
We added exponents (Product Rule). We subtracted exponents (Quotient Rule). Now, we have . This is a power raised to a power.
Past Knowledge
Lesson 8.1. You could just write and add them up, but we want a shortcut.
Today's Goal
Simplify by multiplying the exponents.
Future Success
Essential for Algebraic Manipulation in Calculus.
Key Concepts
The Rule
If you raise a power to a power, MULTIPLY the exponents.
Why?
The "Distributive" Power
If you have multiple things inside the parentheses, the power goes to EVERYTHING.
Don't forget to power up the coefficient too!
Worked Examples
Example 1: Basic Power
BasicSimplify:
Multiply
Answer
Example 2: With a Coefficient
IntermediateSimplify:
Power to Number
The 2 gets the power of 4 too.
Power to Variable
Answer
Example 3: Everything at Once
AdvancedSimplify:
Number
Negative squared becomes positive.
Variables
Multiply powers.
Answer
Common Pitfalls
Adding Instead of Multiplying
With , students sometimes write . Remember the parentheses mean "groups of". 3 groups of 2 is 6.
Forgetting the Coefficient
In , the most common wrong answer is . You must square the 3 also! Correct answer: .
Real-Life Applications
Volume Calculations:
- The volume of a cube is .
- If you double the side length (), the volume becomes .
- The power rule tells us that doubling the size increases the volume by 8 times!
Practice Quiz
Loading...