Math · Radicals
Root
Calculator
Calculate square root, cube root, and any nth root. Solve for the root, original number, or root degree. Includes simplified radical form, all-roots comparison, and perfect squares table.
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ⁿ√x Result
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√x
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∛x
Solve ⁿ√x = r
n√x
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n
Quick
x
r
💡 Negative values: even roots of negatives are undefined in reals. Odd roots (n=3,5…) of negatives are supported.
Root Result
²√144
12
principal root
Verified: rⁿ
144
Negative root
−12
ⁿ√x
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√x
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∛x
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x²
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📊 All Root Degrees — Same x Value
✨ Simplified Radical Form
Simplification
√144
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12
📈 Root Curves: √x, ∛x, ∜x, ⁿ√x
📋 Reference Tables — Click to Load
📐 Root & Radical Rules
Definition
ⁿ√x = x^(1/n)
√x = x^(1/2), ∛x = x^(1/3)
Product Rule
√(ab) = √a × √b
√12 = √4 × √3 = 2√3
Quotient Rule
√(a/b) = √a / √b
√(9/4) = √9/√4 = 3/2
Power Rule
ⁿ√(xᵐ) = x^(m/n)
∛(x²) = x^(2/3)
Nested Roots
ᵐ√(ⁿ√x) = ᵐⁿ√x
√(∛x) = ⁶√x = x^(1/6)
Rationalize
1/√n = √n/n
1/√2 = √2/2 ≈ 0.7071
Even Root (x < 0)
√(negative) = undefined
√(-4) = 2i (complex only)
Odd Root (x < 0)
∛(negative) is real
∛(-8) = -2 ✓
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. The square root (n=2) is the most common, but cube roots (n=3), fourth roots (n=4), and general nth roots all have important applications in mathematics and science.
Key Root Formulas
Find root: r = ⁿ√x = x^(1/n)
Find number: x = rⁿ
Find degree: n = ln(x) / ln(r)
Simplify √n: factor out largest perfect square
√72 = √(36×2) = 6√2
Negative roots: even n → undefined for x < 0
odd n → real result (e.g. ∛(-27) = -3)
When to Use Each Root Type
Square root (√): area to side length, Pythagorean theorem, standard deviation, quadratic formula. Cube root (∛): volume to side length, inverse of cubing, some physics formulas. nth root: compound interest (finding rate), geometric mean of n values, nth degree polynomial roots.
Why does √4 have two answers (2 and −2)?
Both 2² = 4 and (−2)² = 4, so both 2 and −2 are technically square roots of 4. The symbol √ refers specifically to the principal (positive) root. When solving equations like x² = 4, we write x = ±√4 = ±2 to include both. In calculator output, √4 = 2 (principal root only). The negative root is −√4 = −2.
How do you simplify √48?
Find the largest perfect square factor of 48. Factors: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48. Perfect squares among these: 1, 4, 16. Largest is 16. So √48 = √(16 × 3) = √16 × √3 = 4√3. Check: (4√3)² = 16 × 3 = 48 ✓. Always look for the LARGEST perfect square factor to fully simplify in one step.
Is √2 rational or irrational?
√2 is irrational — it cannot be expressed as a fraction p/q where p and q are integers. Its decimal expansion is 1.41421356... continuing without repeating. This was proven by the ancient Greeks (famously attributed to Hippasus). Any square root of a non-perfect-square integer is irrational. Perfect square roots like √4=2 and √9=3 are rational.
What is a perfect square and a perfect cube?
A perfect square is an integer whose square root is also an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100... (n²). A perfect cube is an integer whose cube root is an integer: 1, 8, 27, 64, 125, 216, 343... (n³). Numbers like 64 are both perfect squares (8²) and perfect cubes (4³).