Power Play · Lesson 3 of 7
The Other Side of Powers
“Extend powers to division, zero exponents and negative exponents.”
• Divide powers with the same base using cancellation. • Explain why a non-zero number to power 0 equals 1. • Interpret negative exponents as reciprocals. • Move through positive, zero and negative powers by repeated division. • Use a power line to compare powers.
Division of powers can be understood by writing the powers as repeated factors and cancelling equal factors above and below.
Problem
Simplify 2¹⁰⁰ ÷ 2²⁵.
- 1.The bases are the same.
- 2.Subtract the exponents: 100 − 25 = 75.
- 3.So 2¹⁰⁰ ÷ 2²⁵ = 2⁷⁵.
The base cannot be 0 because division by 0 is not defined. This restriction matters when we extend powers to exponent 0.
When Zero is in Power!
For any non-zero x, xᵃ ÷ xᵃ = 1. The exponent rule gives xᵃ⁻ᵃ = x⁰. Therefore x⁰ = 1 for every non-zero base.
5⁰ = 1, but 0⁵ = 0. Zero exponent and zero base are different ideas.
Continue the same division pattern below exponent 0. Since 2⁰ = 1, one more division by 2 gives 2⁻¹ = 1/2. Another gives 2⁻² = 1/4.
Problem
Write 10⁻⁵ without a negative exponent.
- 1.Use n⁻ᵃ = 1/nᵃ.
- 2.10⁻⁵ = 1/10⁵.
- 3.So 10⁻⁵ = 1/100000.
Problem
Simplify 2⁻⁴ × 2⁷.
- 1.The bases are the same.
- 2.Add exponents: −4 + 7 = 3.
- 3.Therefore, 2⁻⁴ × 2⁷ = 2³ = 8.
Power Lines
A power line arranges consecutive powers of one base. Moving toward a smaller exponent divides by the base; moving toward a larger exponent multiplies by the base.
Problem
How many times larger is 4² than 4⁻²?
- 1.Form 4² ÷ 4⁻².
- 2.Subtract exponents: 2 − (−2) = 4.
- 3.4⁴ = 256.
- 4.So 4² is 256 times as large as 4⁻².
Quiz
Simplify 7⁹ ÷ 7⁴.
What is 13⁰?
Which equals 5⁻³?
Simplify p³ × p⁻¹⁰.
Lowering the exponent by 1 on a base-7 power line does what?
Practice Problems
- Simplify 3¹² ÷ 3⁷ and explain the cancellation.
- Evaluate 6⁰ and explain why it is not 0.
- Write 2⁻⁶ and (−5)⁻³ as fractions.
- Simplify 8⁴ × 8⁻⁹ and write the answer using a positive exponent in the denominator.
- Create a power line from 5² to 5⁻² and write every value.
Key Takeaways
• Same-base division subtracts exponents. • The rule comes from cancelling equal factors. • Any non-zero number to power 0 equals 1. • A negative exponent means reciprocal. • Positive, zero and negative powers form one continuous multiplication/division pattern. • A power line makes this pattern visible.