← Back to the Square Roots And Cube Roots worksheet
Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Standards: 8.EE.A.2 · Level: Grade 8
Problem 1 Evaluate square roots
(a) 8² = 64. (b) 11² = 121. (c) 0.3² = 0.09. (d) 25 squared is 425.
(a) 8 (b) 11 (c) 0.3 (d) 25
Problem 2 Evaluate cube roots
(a) 3³ = 27. (b) 5³ = 125. (c) (−2)³ = −8. (d) (12)³ = 18.
(a) 3 (b) 5 (c) −2 (d) 12
Problem 3 Spot the perfect squares
9² = 81, 10² = 100 and 14² = 196. The number 48 is between 36 and 49, and 150 is between 144 and 169.
81, 100 and 196
Problem 4 Spot the perfect cubes
4³ = 64, 6³ = 216 and 10³ = 1,000. The number 9 is between 8 and 27, and 100 is between 64 and 125.
64, 216 and 1,000
Problem 5 Two small equations
(a) Both 7 and −7 square to 49. (b) Only 5 cubes to 125.
(a) x = 7 or x = −7 (b) x = 5
Problem 6 Patio side length
Side = √169. Since 13 · 13 = 169, the side is 13 feet.
13 feet
Problem 7 Fractions and decimals
(a) 910 squared is 81100, so x = ±910. (b) 0.4² = 0.16, so x = ±0.4.
(a) ±910 (b) ±0.4
Problem 8 Negative and fractional cubes
(a) (−4)³ = −64. (b) 343 = 7³ and 27 = 3³, so x = 37.
(a) x = −4 (b) x = 37
Problem 9 A root that is not whole
x = ±√50. Since 7² = 49, √50 is just above 7. Testing 7.07² = 49.9849 and 7.08² = 50.1264, so √50 ≈ 7.07.
x = ±√50 ≈ ±7.07
Problem 10 A cube root estimate
x = ∛30. Since 3³ = 27 and 4³ = 64, the root is between 3 and 4, and very near 3. Testing 3.1³ = 29.791 and 3.2³ = 32.768, so ∛30 ≈ 3.1.
x = ∛30 ≈ 3.1
Problem 11 Cube-shaped box
Edge = ∛216. Since 6 · 6 · 6 = 216, the edge is 6 inches.
6 inches
Problem 12 Two-step equations
(a) Divide by 2: x² = 49, so x = ±7. (b) Subtract 5: x³ = 64, so x = 4.
(a) x = ±7 (b) x = 4
Problem 13 Estimate two roots
(a) 14² = 196 and 15² = 225. Then 14.1² = 198.81 and 14.2² = 201.64, so √200 ≈ 14.1. (b) 4³ = 64 and 5³ = 125. Then 4.6³ = 97.336 and 4.7³ = 103.823, so ∛100 ≈ 4.6.
(a) 14.1 (b) 4.6
Problem 14 Solutions on a number line
The solutions are x = ±√20. Since 4² = 16 and 5² = 25, they sit between 4 and 5 on each side of zero. 4.4² = 19.36 and 4.5² = 20.25, so √20 ≈ 4.5.
±√20 ≈ ±4.5
Problem 15 Mural frame
Side = √289 = 17 ft because 17² = 289. Perimeter = 4 · 17 = 68 ft. Cost = 68 · $3.25 = $221.00.
17 ft; 68 ft; $221.00
Problem 16 Storage cube surface
Edge = ∛343 = 7 cm, since 7³ = 343. A cube has 6 faces of area 7² = 49 cm². Surface area = 6 · 49 = 294 cm².
Edge 7 cm; surface area 294 cm²
Problem 17 Garden fence
Side = √90. Since 9² = 81 and 10² = 100, and 9.5² = 90.25, the side is just under 9.5: √90 ≈ 9.487. Perimeter = 4 · 9.487 ≈ 37.9 ft.
About 37.9 feet
Problem 18 Why is √2 irrational?
1.41² = 1.9881 and 1.42² = 2.0164, and 1.9881 < 2 < 2.0164, so 1.41 < √2 < 1.42. Every finer decimal also misses 2 exactly, because √2 has a decimal that never ends and never repeats, so no fraction of integers squares to 2.
1.41 < √2 < 1.42; √2 is irrational.
Problem 19 Two cubes
Cube B has edge ∛300. Since 6³ = 216 and 7³ = 343, the edge is between 6 and 7. 6.7³ = 300.763, so ∛300 ≈ 6.7. Difference: 6.7 − 6 = 0.7.
Cube B, by about 0.7 inch
Problem 20 Check a claim
For x³ = −64: (−4)³ = −64, so −4 works. For x² = −16: (−4)² = 16, not −16, and a square of a real number is never negative. So there is no real solution.
Right for x³ = −64; wrong for x² = −16 (no real solution).