Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for square roots and cube roots 8th grade? Then this Grade 8 lesson on Solving With Square and Cube Roots explains each idea step by step, with worked examples.
Quick answer: To solve x² = p, take the square root of both sides: the answers are √p and −√p. To solve x³ = p, take the cube root: there is one answer. Perfect squares and cubes give whole-number roots, while roots such as √2 are irrational and are estimated.
Standards: 8.EE.A.2 · Level: Grade 8
What are square roots and cube roots?
The square root √p is the nonnegative number whose square is p. The cube root ∛p is the number whose cube is p. A perfect square or perfect cube has a whole-number root.
| Number n | n² | n³ |
|---|---|---|
| 1 | 1 | 1 |
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 5 | 25 | 125 |
| 6 | 36 | 216 |
| 7 | 49 | 343 |
| 8 | 64 | 512 |
| 9 | 81 | 729 |
| 10 | 100 | 1000 |
| 11 | 121 | — |
| 12 | 144 | — |
How do you solve x² = p and x³ = p?
x² = p has two solutions, x = √p and x = −√p. x³ = p has exactly one solution, x = ∛p, and it keeps the sign of p.
- Isolate the power on one side of the equation.
- Take the square root (two answers) or the cube root (one answer).
- If the root is not a whole number, leave it exact and estimate between perfect squares or cubes.
- In a measurement problem, keep only the positive length.
A square sticker covers 52 cm². Its side s satisfies s² = 52.
Since 7² = 49 and 8² = 64, 7 < √52 < 8. Testing 7.2² = 51.84 and 7.3² = 53.29 shows the side is about 7.2 cm.
How do cube roots give an edge length?
The volume of a cube is V = e³, so the edge is e = ∛V. A cube holding 64 in³ has edge 4 in, because 4 · 4 · 4 = 64.
Roots such as √2 and ∛30 are irrational: their decimals never end or repeat, so we write the exact root or an approximation.
Writing x = 6 for x² = 36 and stopping. Also, x² = −16 has no real solution, because no real number squared is negative; yet x³ = −64 does, namely −4.
Beep! Memorize the squares up to 15 and the cubes up to 10. They are the landmarks for estimating every other root.
- x² = p (p positive) has two solutions: √p and −√p.
- x³ = p has exactly one solution, ∛p.
- A cube with volume V has edge ∛V; a square with area A has side √A.
- Roots of non-perfect squares and cubes are irrational, like √2.
- x² = negative number has no real solution; x³ = negative number does.
Solving With Square and Cube Roots: frequently asked questions
Why does x² = 49 have two solutions?
Both 7 and −7 give 49 when squared, since (−7)(−7) = 49. We write x = ±7. In a length problem only the positive value makes sense.
What is the cube root of a negative number?
It is negative. For example ∛(−125) = −5, because (−5)(−5)(−5) = −125. Cube roots keep the sign of the number.
Is √2 a rational number?
No. √2 is irrational, so its decimal 1.41421356… never ends and never repeats. No fraction of integers squares to exactly 2.
How do you estimate a cube root that is not a whole number?
Find the two perfect cubes around the number. Since 64 < 100 < 125, ∛100 is between 4 and 5. Test tenths, such as 4.6³ = 97.336, to refine.
How do you find the side of a square from its area?
Take the square root of the area. A square with area 169 square feet has side √169 = 13 feet. Use the positive root only.