A free printable worksheet that pairs with the Exponent Hoops game. The worksheet prints first, and the answer key prints on its own page after it, so you can leave it off. To make a PDF, click Print and choose "Save as PDF."

Exponent Rules with Variables

Grade 8, Algebra 1 · CCSS 8.EE.A.1 · Pairs with the Exponent Hoops game

Name

Date

The rules work for letters just like numbers. Same base: x^a · x^b = x^a+b, x^a ÷ x^b = x^a−b, (x^a)^b = x^ab, x⁰ = 1, x⁻ⁿ = 1 / xⁿ. Write answers with positive exponents.

Part A: Multiplying powers (2-pointers)

1. x³ · x⁵

2. y² · y⁷

3. a · a⁴

4. b⁶ · b⁰

Part B: Dividing powers

5. x⁹ ÷ x⁴

6. y⁷ ÷ y²

7. a⁵ ÷ a⁵

8. b³ ÷ b⁶

Part C: Power of a power (3-pointers)

9. (x²)³

10. (y⁴)²

11. (a⁻²)³

12. (b⁵)⁰

Part D: Put the rules together

13. x⁴ · x² ÷ x⁷

14. (y³)² · y⁻⁴

15. x² · y³ · x⁵ · y

16. Find the mistake. Taylor wrote x³ · x⁵ = x¹⁵. What rule did Taylor use by mistake? What's the correct answer?

Correct:

Answer Key: Exponent Rules with Variables

For the teacher

Add exponents: x⁸
y⁹
a¹ · a⁴ → a⁵
b⁶ · 1 → b⁶
Subtract exponents: x⁵
y⁵
a⁰ → 1
b⁻³ → 1/b³
Multiply exponents: x⁶
y⁸
a⁻⁶ → 1/a⁶
b⁰ → 1
x^(4+2−7) = x⁻¹ → 1/x
y⁶ · y⁻⁴ → y²
Group like bases: x²·x⁵ · y³·y¹ → x⁷y⁴
Taylor multiplied the exponents (the power-of-a-power rule). When multiplying powers, add them: x⁸

A quick proof for #16: write x³ · x⁵ as (x·x·x)(x·x·x·x·x) and count eight x's. In Exponent Hoops, each shot shows the rule used ("Add exponents"), so students hear the rule every time.