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Exponents and Roots unit pack (Grades 6-8 · CCSS 6.EE.A.1, 8.EE.A.1, 8.EE.A.2, 8.NS.A.2): every worksheet and exit ticket for this unit, ready to print in one go. The answer keys print at the end, each on its own page, so you can leave them off. To make a PDF, click Print and choose "Save as PDF." Games: Exponent Racer · Exponent Hoops · Rooted to the Spot

Exponent Rules

Grades 6–8 · CCSS 6.EE.A.1, 8.EE.A.1 · Pairs with the Exponent Racer game
Name
Date
In 53, 5 is the base and 3 is the exponent: 53 = 5 × 5 × 5. Same base: multiply → add the exponents, divide → subtract them. Power of a power → multiply them. a0 = 1, and a−n = 1 / an.

Key vocabulary

Part A: Evaluate each power

1. 25
=
2. 34
=
3. 103
=
4. 70
=
5. 51
=
6. 43
=

Part B: Write as a single power

7. 32 × 34
=
8. 57 ÷ 53
=
9. (23)2
=
10. 108 ÷ 102
=
11. (72)4
=
12. 23 × 25 ÷ 24
=

Part C: Zero and negative exponents (write as a fraction or whole number)

13. 5−2
=
14. 2−3
=
15. 32 × 3−5
=
16. 94 ÷ 94
=

Part D: Vocabulary and reasoning

17. In 64, name the base and the exponent, and find the value of the power.
base exponent value
18. Find the mistake. Sam wrote 43 = 12. What did Sam do wrong? What is the correct value?
Correct:
19. The distance from Earth to the Moon is about 4 × 105 km. Write 105 as a whole number, then write the distance as a whole number.
105 =   distance ≈ km
Play the game: studentmathgames.com/exponent-racer© 2026 Mr. H · Free for classroom use

Exit Ticket: Exponent Rules

Grades 6–8 · CCSS 8.EE.A.1
Name
1. Simplify, then evaluate. Name each rule you used.
(23)2 × 2−4
=
2. Is 10−3 greater than 0? Less than 1? Write it as a decimal.
=
3. Explain why 65 ÷ 65 = 60, and what 60 equals.
60 =

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: xa · xb = xa+b,   xa ÷ xb = xa−b,   (xa)b = xab,   x0 = 1,   x−n = 1 / xn.  Write answers with positive exponents.

Key vocabulary

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:
Play the game: studentmathgames.com/exponent-hoops© 2026 Mr. H · Free for classroom use

Exit Ticket: Exponent Rules with Variables

Grade 8 · CCSS 8.EE.A.1
Name
1. Simplify. a3 · a5 ÷ a2
=
2. Simplify. Use a positive exponent. (y2)−3
=
3. Why can't x2 · y3 be written as a single power?

Penalty Kick Roots

Grade 8 · CCSS 8.EE.A.2, 8.NS.A.2 · Pairs with the Rooted to the Spot game
Name
Date
A square with an area of 49 has sides of 7, because 7 × 7 = 49. So √49 = 7.  ·  ∛8 = 2 because 2 × 2 × 2 = 8.

Key vocabulary

Part A: Squares and square roots

1. 6² = ?
2. 12² = ?
3. √64 = ?
4. √169 = ?

Part B: Side of a square

5. A square rug has an area of 81 square feet. How long is each side?
feet
6. A square field has an area of 225 square meters. How long is each side?
meters

Part C: Estimate and cube roots

7. √30 is between which two whole numbers?
and
8. √90 is between which two whole numbers?
and
9. ∛27 = ?
10. ∛1000 = ?
11. Closer to which? √20 is between 4 and 5. Is it closer to 4 or to 5? Explain how you know.
12. Find the mistake. Jordan said √36 = 18 because 36 ÷ 2 = 18. What went wrong? What is √36 really?
Correct: √36 =
Play the game: studentmathgames.com/rooted-to-the-spot© 2026 Mr. H · Free for classroom use

Exit Ticket: Square Roots and Cube Roots

Grade 8 · CCSS 8.EE.A.2, 8.NS.A.2
Name
1. A square has an area of 121 square units. How long is each side?
2. √60 is between which two whole numbers?
and
3. ∛64 = ?

Answer Key: Exponent Rules

For the teacher
  1. 2·2·2·2·2 = 32
  2. 3·3·3·3 = 81
  3. 1,000
  4. 1 (any nonzero base to the zero power)
  5. 5
  6. 4·4·4 = 64
  7. Add exponents: 36
  8. Subtract exponents: 54
  9. Multiply exponents: 26 (= 64)
  10. 106
  11. 78
  12. 2^(3+5−4) = 24 (= 16)
  13. 1 / 5² = 1/25
  14. 1 / 2³ = 1/8
  15. 3^(2−5) = 3⁻³ = 1 / 3³ = 1/27
  16. 9⁰ = 1
  17. Base 6, exponent 4, value 6·6·6·6 = 1,296
  18. Sam multiplied the base by the exponent (4 × 3). Correct: 4·4·4 = 64
  19. 105 = 100,000; the distance is about 400,000 km

Common mistakes to watch for: multiplying base × exponent (#18), multiplying exponents in the product rule (38 for #7), and reading a negative exponent as a negative number (−25 for #13). Exponent Racer's wrong answer choices are built from these same mistakes.

Full Teacher Guide: studentmathgames.com/exponent-racer/guide© 2026 Mr. H

Answer Key: Exit Ticket

Exponent Rules
  1. Power of a power: (2³)² = 2⁶. Product rule: 2⁶ × 2⁻⁴ = 2² → 4
  2. 10⁻³ = 1 / 10³ = 1/1000 = 0.001. Yes, it's greater than 0; yes, it's less than 1.
  3. Dividing with the same base subtracts exponents: 5 − 5 = 0. Any number divided by itself is 1, so 6⁰ = 1.

Quick read: an answer of 2 on #1 means the student added the exponents in (2³)² and got 2⁵ instead of 2⁶; a negative answer on #2 means they read the exponent as a negative number. Assign the matching skill in Exponent Racer.

Answer Key: Exponent Rules with Variables

For the teacher
  1. Add exponents: x8
  2. y9
  3. a¹ · a⁴ → a5
  4. b⁶ · 1 → b6
  5. Subtract exponents: x5
  6. y5
  7. a⁰ → 1
  8. b⁻³ → 1/b3
  9. Multiply exponents: x6
  10. y8
  11. a⁻⁶ → 1/a6
  12. b⁰ → 1
  13. x^(4+2−7) = x⁻¹ → 1/x
  14. y⁶ · y⁻⁴ → y2
  15. Group like bases: x²·x⁵ · y³·y¹ → x7y4
  16. 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.

Full Teacher Guide: studentmathgames.com/exponent-hoops/guide© 2026 Mr. H

Answer Key: Exit Ticket

Exponent Rules with Variables
  1. a^(3+5−2) → a6
  2. y^(2·−3) = y⁻⁶ → 1/y6
  3. The rules only combine powers with the same base. x and y are different bases, so the exponents can't be added.

Quick read: a13 or a15 on #1 means the student multiplied the exponents; y−1 or y5 on #2 means they added in a power of a power.

Answer Key: Penalty Kick Roots

For the teacher
  1. 6 × 6 = 36
  2. 12 × 12 = 144
  3. 8 (8 × 8 = 64)
  4. 13 (13 × 13 = 169)
  5. 9 feet (9 × 9 = 81)
  6. 15 meters (15 × 15 = 225)
  7. 5 and 6 (25 < 30 < 36)
  8. 9 and 10 (81 < 90 < 100)
  9. 3 (3 × 3 × 3 = 27)
  10. 10 (10 × 10 × 10 = 1000)
  11. Between 4 and 5 (16 < 20 < 25). Closer to 4: 20 is 4 away from 16 but 5 away from 25. (√20 ≈ 4.47)
  12. Jordan divided by 2 instead of finding the number that times itself makes 36. 6 × 6 = 36, so √36 = 6

#12 is the most common mistake: halving instead of undoing the square. In the game, a wrong answer draws the square as a grid so students can see that 6 rows of 6 make 36.

Full Teacher Guide: studentmathgames.com/rooted-to-the-spot/guide© 2026 Mr. H

Answer Key: Exit Ticket

Square Roots and Cube Roots
  1. 11 (11 × 11 = 121)
  2. 7 and 8 (49 < 60 < 64)
  3. 4 (4 × 4 × 4 = 64)

Quick read: 60.5 on #1 means the student halved the area; 21 on #3 means they divided by 3. Have them play with just Square Roots and Side of a Square turned on, where each wrong answer draws the square, then add Cube Roots.