What is the Cube Root of 8?
The exact value of the Cube Root of 8, plus a calculator to solve for the degree or radicand instead.
Solved for Result
2
∛8 = 2
Degree
3
Radicand
8
Result
2
What is a Root Calculator?
A root calculator finds the number that, multiplied by itself a set number of times, produces a given value. The most familiar case is the square root (√a): the number that, squared, gives a. Cube roots (∛a) and higher nth roots work the same way, just with more repeated multiplications. This calculator also works backward — given the root's degree and result, it finds the radicand, or given the radicand and result, it finds the degree.
Roots are the inverse operation of exponents: taking the nth root of a is the same as raising a to the power 1/n. That's exactly how this calculator computes results internally, and why the Exponent and Root calculators are closely related tools.
Root Formula
Here, n is the root's degree (2 for square root, 3 for cube root, and so on), a is the radicand (the number under the root symbol), and b is the result.
Why Negative Numbers Behave Differently
Square roots (and any even-degree root) of a negative number have no real-number answer — no real number squared produces a negative result, since a negative times a negative is always positive. Cube roots (and any odd-degree root) of a negative number do have a real answer, though: ∛(−8) = −2, because (−2)³ = −8. This calculator allows negative radicands only when the root degree is an odd integer, and flags other combinations as undefined rather than guessing.
The Babylonian (Long-Division) Method for Estimating Square Roots
Before calculators, square roots were commonly estimated by hand using an iterative averaging method: start with a rough guess, divide the target number by that guess, then average the guess and the result of that division to get a better guess. Repeating this a few times converges quickly on the true square root — for example, estimating √50: start with a guess of 7 (49 is close to 50); 50 ÷ 7 ≈ 7.14; average of 7 and 7.14 is 7.07 — already within 0.001 of the true value (√50 ≈ 7.0711). The same averaging idea extends to nth roots with a slightly modified formula.
Example — Your Current Inputs
∛8 = 2
Additional Example — The Pythagorean Theorem
Square roots show up constantly in geometry. For a right triangle with legs of 6 and 8, the hypotenuse is √(6² + 8²) = √(36 + 64) = √100 = 10. Roots are how you go from a sum of squared lengths back to an actual distance — the same operation underlies distance formulas in coordinate geometry, physics, and statistics (like the root-mean-square and standard deviation).
About These Parameters
- Root Degree (n)
- Which root you're taking. 2 is the square root (by far the most common), 3 is the cube root, and any positive integer beyond that is a valid "nth root."
- Radicand
- The number under the radical symbol — the value you're finding the root of. Negative radicands are only valid with an odd root degree.
- Result
- The value that, raised to the root degree, equals the radicand. Leave blank to solve for it directly; fill it in with either degree or radicand to solve for the other.
Frequently Asked Questions
Is every square root either a whole number or irrational?
Yes, for whole-number radicands. If a positive integer isn't a "perfect square" (1, 4, 9, 16, 25...), its square root is always an irrational number — a never-repeating, never-ending decimal. There's no such thing as a square root of a non-perfect-square integer that comes out to a "nice" fraction.
What's the difference between a root and an exponent?
They're inverse operations. An exponent tells you how many times to multiply a base by itself (2³ = 8). A root asks the reverse question: what number, multiplied by itself n times, gives this result? (∛8 = 2). In fact, a root is just a fractional exponent: ⁿ√a = a^(1/n).
Does every positive number have two square roots?
Yes — both a positive and a negative number, when squared, give the same positive result. For example, both 5 and −5 squared equal 25. By convention, the "√" symbol refers only to the positive (principal) square root, but when solving equations like x² = 25, both roots (x = 5 and x = −5) are valid solutions.
Can the root degree be a decimal?
Mathematically yes — since ⁿ√a = a^(1/n), any positive real n is valid as long as the radicand is positive. A degree of 2.5, for example, is a perfectly well-defined operation on a positive radicand, just less common in everyday use than whole-number roots.