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Mastering Place Value: Whole Numbers for Ontario Grade 6 Math
Lesson 3: Zero and Negative Exponents
Exploring the patterns behind zero and negative powers
Welcome to Lesson 3! So far, you’ve learned how to work with positive exponents and apply the laws of exponents for multiplication and division. But what happens when we encounter an exponent of zero or a negative exponent? These might seem strange at first, but they follow logical patterns that make perfect sense once you understand them.
In this lesson, we’ll explore:
- What zero exponents mean and why any non-zero number to the power of zero equals 1
- What negative exponents represent
- How to convert between negative exponents and fractions
- How to apply the laws of exponents with zero and negative exponents
Understanding Zero Exponents
Question: What is 5⁰?
Before we answer, let’s use the division law to discover the pattern.
Remember the division law: When dividing powers with the same base, we subtract the exponents.
Let’s calculate 5³ ÷ 5³ in two different ways:
Method 1: Calculate First
Method 2: Division Law
Both methods give us the same result, so: 5⁰ = 1
📐 THE ZERO EXPONENT RULE
For any non-zero number a:
a⁰ = 1
Important note: 0⁰ is undefined in mathematics. But any other number to the power of zero equals 1.
Why Does This Make Sense?
Let’s look at a pattern of decreasing exponents:
Notice that each time we decrease the exponent by 1, we divide the result by 2. Following this pattern: 2¹ = 2, so 2⁰ = 2 ÷ 2 = 1
This pattern holds for any base!
Examples: Zero Exponents
Example 1: Evaluate 10⁰
Solution: 10⁰ = 1
Example 2: Evaluate (-7)⁰
Solution: (-7)⁰ = 1
(Any non-zero number, even negative ones, to the power of zero equals 1)
Example 3: Evaluate 3 × 4⁰
Solution:
First, evaluate 4⁰ = 1
Then: 3 × 1 = 3
Answer: 3
Understanding Negative Exponents
Question: What is 2⁻³?
Let’s use the pattern approach again:
Following the pattern: 2⁻¹ = 1 ÷ 2 = 1/2
Continuing: 2⁻² = 1/2 ÷ 2 = 1/4 and 2⁻³ = 1/4 ÷ 2 = 1/8
Notice that 1/8 can be written as 1/2³
📐 THE NEGATIVE EXPONENT RULE
For any non-zero number a and positive integer n:
a⁻ⁿ = 1/aⁿ
In words: A negative exponent means “take the reciprocal and make the exponent positive.”
Why Does This Make Sense with the Division Law?
Using the division law:
2² ÷ 2⁵ = 22-5 = 2⁻³
But we can also write this as:
2² ÷ 2⁵ = 4 ÷ 32 = 4/32 = 1/8
And 1/8 = 1/2³
So: 2⁻³ = 1/2³
The negative exponent rule is consistent with the division law!
Examples: Negative Exponents
Example 1: Evaluate 3⁻²
Solution:
3⁻² = 1/3² = 1/9
Answer: 1/9
Example 2: Evaluate 5⁻¹
Solution:
5⁻¹ = 1/5¹ = 1/5
Answer: 1/5
Example 3: Evaluate 10⁻⁴
Solution:
10⁻⁴ = 1/10⁴ = 1/10,000 = 0.0001
Answer: 1/10,000 or 0.0001
Example 4: Rewrite 1/4³ using a negative exponent
Solution:
1/4³ = 4⁻³
Answer: 4⁻³
This works in reverse too! If you have a fraction with 1 in the numerator, you can write it as a negative exponent.
Negative Exponents in the Denominator
What about expressions like 1/3⁻²?
When the negative exponent is in the denominator, we can flip it to the numerator and make it positive:
Shortcut: 1/a⁻ⁿ = aⁿ
General Rule:
- a⁻ⁿ = 1/aⁿ (negative exponent in numerator → flip to denominator and make positive)
- 1/a⁻ⁿ = aⁿ (negative exponent in denominator → flip to numerator and make positive)
Combining Zero and Negative Exponents with Laws of Exponents
All the laws we learned in Lesson 2 still apply with zero and negative exponents!
Example 1: Simplify 3⁵ × 3⁻²
Solution:
Using the multiplication law: 3⁵ × 3⁻² = 35+(-2) = 3³
Answer: 3³ or 27
Example 2: Simplify 5⁻³ ÷ 5⁻⁷
Solution:
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What You’ll Learn?
Course Syllabus
Introduction to Place Value: Understanding the Basics
Exploring Large Numbers: Reading and Writing Whole Numbers
Comparing and Ordering: Strategies for Whole Numbers
Advanced Place Value: Rounding and Estimation Techniques
Mastery and Application: Problem Solving with Place Value
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