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Root Calculator

Calculate square, cube, or nth roots.

Root Lab

Enter Degree (n) and Radicand (a) to find the nth root. Or enter Degree and Result to solve for the radicand (Inverse Mode). Calculated fields use a dashed mint border.

Formatting

Round to…
Sig figs
Results

Root · Perfect

4

⁽2⁾√16 = 4

Verification

b2 = 42 ≈ 16

Logic trace

  • Exponential form⁽2⁾√16 = 16^{1/2}
  • Result⁽2⁾√16 = 4

Estimation (Newton's method)

xnew = ((n−1)·x + a/xn−1) / n, with n = 2

IterGuessNext
185
254.1
34.14.0012

How to Use the Root Lab

Enter Degree (n) and Radicand (a) to find ⁿ√a, or use Inverse Mode to solve for the radicand. The Logic Trace shows the formula; the Estimation Table illustrates Newton's method. All calculations run locally.

Quick Tips

Presets

√ Square Root and ∛ Cube Root set degree and sample radicands. General nth roots: enter any positive integer degree.

Inverse Mode

Know the root? Enter Degree + Result, leave Radicand blank to get a = bⁿ.

Verification

The dashboard shows bⁿ ≈ a. Use Copy results and format controls (rounding, sig figs) for export.

Root Calculator: Square, Cube & Nth Root Solver

Free square root calculator, cube root calculator, and nth root solver. How to find square root and nth root formula with steps. Find ⁿ√a or solve for radicand (inverse). Newton's method, verification bⁿ ≈ a. Trusted by students. No sign-up.

What This Root Calculator Does & Who It's For

Calculator Purpose & Ideal Users

This root calculator computes nth roots (ⁿ√a = b) and supports square root, cube root, and general roots (4th, 5th, 8th, etc.). Inverse mode solves for the radicand when you know the root.
  • What You'll Get:
    Root from degree + radicand; inverse: radicand from degree + result. Perfect Root highlight, verification bⁿ ≈ a, Logic Trace (exponential form a^(1/n)), Newton's method Estimation table (3 iterations), Copy results, rounding/sig figs.
  • Ideal Users:
    Students: How to find square root, cube root, nth root formula. Teachers: Newton's method demo, exponential form. Anyone: Quick square root calculator or 8th root calculator for homework or verification.
  • Scope:
    Degree n: positive integer. Even roots of negative numbers undefined. All calculations local.

Nth Root Formula: ⁿ√a = a^(1/n)

The nth root of a is b where
bn=ab^n = a
In exponential form:
an=a1/n\sqrt[n]{a} = a^{1/n}
Examples: √16 = 4, ∛27 = 3, ⁸√15 ≈ 1.403. The Logic Trace substitutes your values into this form

How to Use This Calculator

Enter a degree and radicand to find a root, or use Inverse Mode to solve for the radicand from a known result.
  • Root Type / Degree:
    Click the √ Square Root or ∛ Cube Root preset, or type any positive integer (4, 5, 8, etc.) in the Degree field for general nth roots.
  • Radicand:
    The number whose root you want. For Inverse Mode, leave this blank and enter the Result instead to solve for a = bn.
  • Result (Inverse Mode):
    Enter the known root value and degree to solve for the radicand. The solved field shows a dashed mint border.
  • Verification & Logic Trace:
    The dashboard confirms bn ≈ a and flags Perfect Roots (exact integers). The Logic Trace shows the exponential form a^(1/n) with your values substituted.
  • Estimation Table:
    Displays the first 3 iterations of Newton's method so you can see how the approximation converges to the answer.

Square Root, Cube Root & General Nth Root

Square root (n=2) and cube root (n=3) have dedicated presets. For nth root calculation with n = 4, 5, 8, etc., enter the degree directly. The radical notation ⁽ⁿ⁾√a updates dynamically.

Inverse Mode: Solve for Radicand

If you know the root (b) and degree (n), use Inverse Mode to find the radicand: a = bⁿ. Example: ⁸√a = 1.5 → a = 1.5⁸ ≈ 25.63. Enter Degree and Result; leave Radicand blank.

Newton's Method: How Nth Roots Are Approximated

The Root Lab uses Newton's method, a standard numerical algorithm with quadratic convergence:
xnew=(n−1)x+a/xn−1nx_{\text{new}} = \frac{(n-1)x + a/x^{n-1}}{n}
The Estimation Table shows the first 3 iterations, useful for understanding how the approximation converges

Root Calculator FAQ

What is an nth root?

The nth root of a is the number b such that bⁿ = a. Notation: ⁿ√a = b. Square root (n=2): √16 = 4. Cube root (n=3): ∛27 = 3. This calculator finds square roots, cube roots, and any general nth root (4th, 5th, 8th, etc.).

How do I find the square root or cube root?

Use the √ Square Root (n=2) or ∛ Cube Root (n=3) presets, or set Degree manually. Enter the Radicand and leave Result blank. The verification card confirms bⁿ ≈ a.

What is Inverse Mode?

When you know the root (b) and degree (n), Inverse Mode solves for the radicand: a = bⁿ. Enter Degree and Result; leave Radicand blank. The solved field shows a dashed mint border.

What does "Perfect Root" mean?

A perfect root is an exact integer result (e.g. √16 = 4, ∛27 = 3). The dashboard highlights "Perfect Root" when the computed value has no decimal part.

What is Newton's method in the Estimation Table?

The calculator uses Newton's method, a standard numerical algorithm for nth roots:
xnew=(n−1)x+a/xn−1nx_{\text{new}} = \frac{(n-1)x + a/x^{n-1}}{n}
The Estimation Table shows the first 3 iterations so you see convergence

Can I calculate roots like the 8th root or 5th root?

Yes. Enter any positive integer degree (4, 5, 8, etc.). A "General n-th Root Mode" badge appears when n > 3. Example: ⁸√15 ≈ 1.403.

Mathematical Reference Note

Calculation Logic: This tool uses standard mathematical algorithms. While we strive for accuracy, errors in logic or user input can result in incorrect data.

Verification: Results should be cross-checked if used for important academic, professional, or personal calculations.

Standard Terms: This tool is provided free of charge and as-is. CalcRegistry provides no warranty regarding the accuracy or fitness of these results for your specific needs.

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