Root Calculator
Calculate square root, cube root, and any nth root. Solve for the root, the original number, or the root degree — with simplified radical form, an all-roots comparison, and a perfect-powers table.
12
ⁿ√x Result
12
√x
5.241483
∛x
◆ Solve ⁿ√x = r
⚠ Negative values: even roots of negatives are undefined in reals. Odd roots (n = 3, 5, 7…) of negatives are supported.
Root Result
12
principal root (√x) = 144
n = 2
144
Verified by rⁿ
±12
Negative Root
12
√x
5.241483
∛x
20736
x²
20736
x2
✓How this checks out
√144 = 12, because 12² = 144
This is a perfect root — the result is a whole number.
All Root Degrees — Same x Value
√
Square Root
12
√144 = 12
³√
Cube Root
5.241483
³√144 = 5.241483
⁴√
4th Root
3.464102
⁴√144 = 3.464102
⁵√
5th Root
2.70192
⁵√144 = 2.70192
¹⁰√
10th Root
1.643752
¹⁰√144 = 1.643752
✦ Simplified Radical Form
Simplification
√144
↓
12
Find the largest perfect square factor: 144 = 144 × 1
Separate using the product rule: √144 = √144 × √1
Simplify the perfect-power part: √144 = 12, so √144 = 12
📈 Root Curves: √x, ∛x, ⁴√x, ⁿ√x
📋 Reference Tables — Click to Load
Root & Radical Rules
Definition
ⁿ√x = x^(1/n)
5² = 25, so √25 = 5
Product Rule
√(ab) = √a · √b
√12 = √4 · √3 = 2√3
Quotient Rule
√(a/b) = √a / √b
√(9/4) = 3/2
Power Rule
ⁿ√(xᵐ) = x^(m/n)
∛(6²) = 6^(2/3)
Nested Roots
ⁿ√(ᵐ√x) = ⁿᵐ√x
√(∛64) = ⁶√64
Rationalize
1/√n = √n / n
1/√2 = √2/2 ≈ 0.7071
Even Root (x < 0)
√(negative) = undefined
√-4 is not real (complex only)
Odd Root (x < 0)
∛(negative) is real
∛-27 = -3
Understanding Roots and Radicals
The nth root of a number x (written ⁿ√x) is the value r such that rⁿ = x. Roots are the inverse operation of powers — where an exponent repeatedly multiplies a number by itself, a root asks what number, multiplied by itself that many times, gets you back to where you started. The square root (n = 2) is the most common case, but cube roots (n = 3), fourth roots (n = 4), and general nth roots all show up regularly in geometry, physics, and finance.
The one nuance worth remembering is what happens with negative radicands. An odd-degree root of a negative number (like ∛-8) is real and negative — real numbers happily multiply out to a negative result an odd number of times. An even-degree root of a negative number (like √-4) has no real solution, because any real number squared, or raised to any even power, is zero or positive. That’s the gap imaginary numbers were invented to fill.
Key Root Formulas
Find root: r = ⁿ√x = x^(1/n)Find number: x = rⁿFind degree: n = ln(x) / ln(r)Simplify √n: factor out the largest perfect square (e.g. √72 = √(36×2) = 6√2)Negative roots: even n → undefined for x < 0. Odd n → real result (e.g. ∛-27 = -3)√
Square Root (n = 2)
Used for side length from area, the Pythagorean theorem, standard deviation, and the quadratic formula. Example: √81 = 9, because 9² = 81.
∛
Cube Root (n = 3)
Used to find side length from volume, and to invert cubing in physics formulas. Example: ∛125 = 5, because 5³ = 125.
ⁿ√
nth Root (any n)
Used for compound interest rate solving and finding the geometric mean of n values. Example: ⁴√81 = 3, because 3⁴ = 81.
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