Rainbow 6: Properties of Exponents
Materials Needed
- Pencil and paper
- Colored pencils, crayons, or highlighters in six colors
- Timer
- Practice worksheet and exit ticket
- Optional: index cards labeled with exponent rules
Learning Objectives
By the end of the lesson, I can:
- Recall the rules for simplifying expressions with exponents.
- Generate an equivalent algebraic expression using at least one property of exponents.
- Explain which exponent property I used.
Success Criteria
I am successful when I can:
- Identify the correct exponent property.
- Apply the property accurately.
- Write an expression that has the same value as the original expression.
- Explain my reasoning using mathematical vocabulary.
Introduction: Rainbow 6 Challenge (2 minutes)
Hook: Imagine you are designing a rainbow-powered game machine. Each color unlocks a different exponent rule. Can you unlock all six colors by creating equivalent expressions?
Today’s goal is to review six common properties of exponents and use at least one property to rewrite an expression without changing its value.
The Six Rainbow Rules
| Color | Property | Rule | Example |
|---|---|---|---|
| Red | Product of Powers | am · an = am+n | x3 · x4 = x7 |
| Orange | Quotient of Powers | am ÷ an = am−n | y8 ÷ y3 = y5 |
| Yellow | Power of a Power | (am)n = amn | (z2)3 = z6 |
| Green | Power of a Product | (ab)n = anbn | (2x)3 = 23x3 |
| Blue | Zero Exponent | a0 = 1, where a ≠ 0 | 50 = 1 |
| Purple | Negative Exponent | a−n = 1/an | x−3 = 1/x3 |
Body
I Do: Teacher Modeling (4 minutes)
Choose one color and model the steps aloud. Use the Red property:
Example: Rewrite a4 · a3 as an equivalent expression.
- Notice that the bases are the same: both bases are a.
- Identify the property: Product of Powers.
- Add the exponents: 4 + 3 = 7.
- Write the equivalent expression: a7.
Think-aloud: “I add the exponents because I am multiplying powers with the same base. I do not multiply the bases or the exponents.”
Model one additional example:
(m2)4 = m8
Here, the exponents are multiplied because this is the Power of a Power property.
Quick Check
Ask: What property would you use to rewrite p5 · p2? Why?
Expected response: Product of Powers; add the exponents to get p7.
You Do: Rainbow 6 Solo Challenge (4 minutes)
Complete the six challenges below independently. After each answer, color or mark the matching rainbow color. You earn one point for each correct answer and one bonus point for naming the property.
- Red: x3 · x5 = ______
- Orange: y9 ÷ y4 = ______
- Yellow: (n3)2 = ______
- Green: (3r)2 = ______
- Blue: q0 = ______
- Purple: t−4 = ______
Self-check: Circle the answer you feel most confident about and put a question mark beside one you would like to discuss.
We Do: Rainbow Repair Game (3 minutes)
Work together to repair two “broken” expressions. The learner explains the mistake and corrects it.
Broken expression 1: b2 · b6 = b12
Discuss: What went wrong? What should the answer be?
Correction: The exponents should be added, not multiplied: b2+6 = b8.
Broken expression 2: (4s)2 = 4s2
Discuss: What part of the expression also needs the exponent?
Correction: The exponent applies to both factors: (4s)2 = 42s2 = 16s2.
Practice Worksheet: Rainbow 6 Exponent Expressions
Directions: Rewrite each expression as an equivalent expression. Name the exponent property you used when requested.
- a4 · a6 = ______________________________
- c11 ÷ c5 = ______________________________
- (h4)3 = ______________________________
- (2k)3 = ______________________________
- w0 = ______________________________
- v−5 = ______________________________
- Rewrite m8 as a product of powers in two different ways.
- Rewrite z12 as a power of a power.
- Identify the property and simplify: (3x)2.
- Real-world problem: A square video display has a side length of 23 meters. Its area is found by squaring the side length: (23)2. Rewrite the expression using the Power of a Power property and find the area.
Worksheet Answer Key
- a10 — Product of Powers
- c6 — Quotient of Powers
- h12 — Power of a Power
- 23k3, or 8k3 — Power of a Product
- 1 — Zero Exponent
- 1/v5 — Negative Exponent
- Possible answers: m3 · m5 and m2 · m6
- (z3)4, or (z2)6
- 9x2 — Power of a Product
- (23)2 = 26 = 64 square meters
Exit Ticket (2 minutes)
Complete independently before ending the lesson.
- Rewrite r3 · r7 as an equivalent expression.
- Name the property you used.
- In one sentence, explain why your new expression is equivalent to the original expression.
Exit ticket answer: r10; Product of Powers. The exponents are added because the bases are the same and the powers are being multiplied.
Conclusion and Reflection
Recap the lesson by asking the learner to complete this sentence:
“To generate an equivalent expression, I first identify the __________, then I apply the matching __________.”
Key takeaway: Equivalent expressions may look different, but they have the same value. The correct exponent rule depends on the structure of the expression.
Differentiation and Extensions
- Support: Keep the rainbow rules visible. Highlight matching bases and circle the operation before choosing a rule.
- Extra support: Use numerical examples first, such as 23 · 22 = 25, before returning to variables.
- Choice: Complete the worksheet on paper, type answers digitally, or explain answers aloud.
- Extension: Create one challenge card for each rainbow color, including an answer and explanation.
- Challenge: Explain why x2 + x3 cannot be simplified using the Product of Powers property.