Square Root & Cube Root Calculator
Square roots, cube roots, and nth roots with steps and Newton's method approximations.
Square Root & Cube Root Finder
Calculate square roots, cube roots, and nth roots with detailed explanations and approximation methods
Quick Examples
Roots and radicals
ⁿ√a = b means bn = a. Square root (n = 2) and cube root (n = 3) are the usual cases; any positive integer degree works.
Even roots need a ≥ 0 in the reals and return non-negative values. Odd roots accept negatives and keep the sign: ∛(-8) = -2.
Square root: Pythagorean theorem, distance, std. deviation
Cube root: volumes, cubic equations
Higher roots: engineering and polynomial work
Properties
Perfect cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
Newton's method for √a
Iterate xn+1 = ½(xn + a/xn). Digits roughly double each step. The Babylonian method is the same idea for square roots.
Uses
- Geometry: c = √(a² + b²); distance √[(x₂−x₁)² + (y₂−y₁)²]
- Physics: pendulum T = 2π√(L/g); escape velocity √(2GM/r)
- Stats: standard deviation is √variance; SE = σ/√n
- Finance: volatility often scales with √t
Common mistakes
- √(a + b) ≠ √a + √b
- Treating √(-4) like -√4
- Assuming √ always returns ±
- Principal square root is ≥ 0
- Check domain before even roots of negatives
- Verify by raising the answer to the nth power
Frequently Asked Questions
Why no real square root of negatives?
No real number squares to a negative. Complex numbers use i = √-1. Odd roots of negatives are fine in the reals.
√4 vs ±√4?
√4 is the principal root, 2. ±√4 means both 2 and -2 (as in x² = 4).
Exact or approximate?
Exact when the result is rational (√16 = 4). Most other roots are irrational and get approximated.
How accurate is Newton's method?
With a decent start, a few iterations usually match calculator precision.
Roots and exponents?
ⁿ√a = a^(1/n). That is why (ⁿ√a)n = a (with the usual caveats for even roots).
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