← Back to the Integer Exponents worksheet
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Standards: 8.EE.A.1 · Level: Grade 8
Problem 1 Repeated multiplication
(a) 4 · 4 · 4 = 64. (b) (−2)(−2)(−2)(−2)(−2): five negative factors give a negative product, −32. (c) 10 · 10 · 10 · 10 = 10,000.
(a) 64 (b) −32 (c) 10,000
Problem 2 Zero and negative exponents
(a) Any nonzero base to the zero power is 1. (b) 2−4 = 124 = 116. (c) 10−3 = 11,000 = 0.001.
(a) 1 (b) 116 (c) 11,000
Problem 3 Product rule
Same base, so add exponents. (a) 5 + 2 = 7. (b) 9 + 4 = 13.
(a) 67 (b) m13
Problem 4 Quotient rule
The bottom has two factors of 6. They cancel two of the five on top, leaving three. Rule check: 5 − 2 = 3.
63 = 216
Problem 5 Power of a power
Two boxes of three 2s make six factors: 3 · 2 = 6.
So (23)2 = 26 = 64. In the same way (n5)2 = n10.
26 = 64 and n10
Problem 6 Counting factors
Six circles in all, so the power is 36. 36 = 3 · 3 · 3 · 3 · 3 · 3 = 729.
36 = 729
Problem 7 Negative exponents as fractions
(a) 4−3 = 143 = 164. (b) The reciprocal of 15 is 5, so (15)−2 = 52 = 25.
(a) 164 (b) 25
Problem 8 Mixed exponents, one base
(a) −2 + 4 = 2, so 52 = 25. (b) 3 + (−5) = −2, so 7−2 = 149.
(a) 25 (b) 149
Problem 9 Power of a product
Apply the exponent to every factor. (a) 53y3 = 125y3. (b) 43m6 = 64m6. (c) 22a6 = 4a6.
(a) 125y3 (b) 64m6 (c) 4a6
Problem 10 Power then divide
(52)3 = 56. Then 56 ÷ 54 = 52 = 25.
25
Problem 11 Fractions to a power
(a) The exponent applies to top and bottom: 2333 = 827. (b) A negative exponent flips the fraction: (43)2 = 169.
(a) 827 (b) 169
Problem 12 Positive exponents only
(a) 1x3. (b) Only a moves: 5a2. (c) A negative exponent in the denominator moves up: y4.
(a) 1x3 (b) 5a2 (c) y4
Problem 13 Complete the pattern
Each box is the previous one divided by 5: 25 ÷ 5 = 5, 5 ÷ 5 = 1, 1 ÷ 5 = 15.
51 = 5, 50 = 1, 5−1 = 15; divide by 5 each step.
Problem 14 Find the error
Zoe multiplied the bases. The base stays 4 and the exponents add: 3 + 2 = 5. Check: 64 · 16 = 1,024, and 165 is far larger.
45 = 1,024
Problem 15 Share chain
Round 6: 36 = 729 students. Round 4: 34 = 81. Ratio: 36 ÷ 34 = 32 = 9, and 729 ÷ 81 = 9 agrees.
729 students; 9 times as many
Problem 16 Nested powers
Inside: −3 + 8 = 5, so (25)2 = 210. Then 210 ÷ 27 = 23 = 8.
8
Problem 17 Simplify with two variables
Multiply each exponent by −2: x4y−6. Move y−6 to the denominator.
x4y6
Problem 18 Compare three values
2−3 = 18. (−2)−3 = 1(−2)3 = −18. −23 means the negative of 8, which is −8.
−23 < (−2)−3 < 2−3
Problem 19 Cookie batches
Total = 2−3 · 26 = 23 = 8. Check: 2−3 = 0.125 cup and 64 · 0.125 = 8.
23 = 8 cups
Problem 20 Missing exponents
(a) n − 4 = 9, so n = 13. (b) 3m = 18, so m = 6. (c) k − (−2) = 7, so k + 2 = 7 and k = 5.
(a) 13 (b) 6 (c) 5