Square / Cube / Nth Root Calculator

Inputs

Expression: √50
Index (root order)

Square root

Answer

Result
5√2 ≈ 7.071068
Simplified radical
5√2
Perfect root?
No
Approximation
7.0710678119

Show your work

Given
  • √50index = 2, radicand = 50
  1. Factor out the largest perfect power
    √50 = √(25 · 2)
    = 5√2
Answer (approximation)
7.07106781
√50 ≈ 7.0711

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Pythagorean Theorem

Solve a right triangle — the final step (√ of a² + b²) is exactly the root operation this calculator performs.

How the nth root calculator works

The nth root of a number x is the value r that satisfies rⁿ = x. For n = 2 it's called the square root, for n = 3 the cube root, and for higher n, the nth root.

This calculator does three things at once:

  1. Checks for a perfect root — if rⁿ exactly equals the input (e.g., √144 = 12, ³√27 = 3, ⁵√32 = 2), returns the exact integer.
  2. Attempts simplified radical form — if the radicand has a perfect nth-power factor (e.g., 50 = 25 · 2 and 25 = 5²), extracts it and rewrites as 5√2. Works for any n: ⁴√80 = 2 · ⁴√5 etc.
  3. Approximates — gives the decimal value to 10 places whether the root is exact, simplified, or plainly irrational (like √2).

Perfect roots at a glance

Simplifying radicals — the quick rule

To simplify ⁿ√a, find the largest integer k such that kⁿ divides a. Then ⁿ√a = k · ⁿ√(a / kⁿ).

Negative radicands — odd vs even index

Odd index — roots of negative numbers are real: ³√−8 = −2, because (−2)³ = −8. Works for any odd n.

Even index — roots of negative numbers are complex. There's no real r with r² = −4, but there is a complex one: the principal square root is 2i (and the other root is −2i). This calculator returns the principal complex root automatically: √−4 = 2i, ⁴√−16 = √2 + √2·i, and so on. The formula for the principal root is |x|^(1/n) · (cos(π/n) + i·sin(π/n)).

How to find an nth root by hand — four classical methods

Pocket calculators compute roots in microseconds, but knowing the methods helps you catch errors and build intuition. These are the techniques every math class teaches.

1. Prime factorization (for perfect and simplifiable roots)

Break the radicand into prime factors, then group them in batches of n. Each complete group of n of the same prime becomes one factor outside the radical; leftovers stay inside. This is exactly how the calculator gets √72 = 6√2:

  72 = 2 · 2 · 2 · 3 · 3
  For √ (index 2), group in pairs:
    (2 · 2) · 2 · (3 · 3)
    →  2 · 3  outside,  2  inside
  Answer: 6√2

2. Estimation between integers (sanity check)

Any irrational root sits between two integers. Find the two nearest perfect nth powers, bracket the root between them, and refine by linear interpolation.

  √50 sits between √49 = 7 and √64 = 8
  → answer is ≈ 7 + (50−49)/(64−49) = 7 + 1/15 ≈ 7.07
  (exact: 7.0710678…)

3. Newton's iterative method (for any n and any radicand)

Newton's method converges on ⁿ√x quickly. Starting from a guess r₀:

  rₖ₊₁ = ((n−1) · rₖ + x / rₖ^(n−1)) / n

For square roots (n = 2), this reduces to the ancient Babylonian method: rₖ₊₁ = (rₖ + x/rₖ) / 2. Each iteration roughly doubles the correct digits, so 3–5 iterations from a rough guess give near-perfect precision. Example for √50 starting from r₀ = 7:

  r₁ = (7 + 50/7) / 2 = (7 + 7.1428) / 2 = 7.0714
  r₂ = (7.0714 + 50/7.0714) / 2 = 7.07106…
  r₃ ≈ 7.0710678   (matches the true value)

4. Logarithms (historical method)

Before calculators, students used log tables: ⁿ√x = 10^(log₁₀ x / n). Look up log x, divide by n, then take the antilog. Slower than Newton's, but conceptually simple. Modern calculators use this same identity internally with natural logs: ⁿ√x = e^(ln x / n).

Common mistakes to avoid