Root Calculator

Free online Root Calculator with step-by-step solutions. 100% client-side, no data upload.

$\sqrt{\phantom{x}}$

√

Radicand

—

Updates live as you type · Ctrl + Enter also works

Recent Calculations

No calculations yet — your last 5 will show up here.

Root Calculator

This free online root calculator computes square roots, cube roots, and general nth roots instantly in your browser, with no signup, no ads, and no server uploads.

What is a root? The $n$-th root of a number $a$ is a value $b$ such that $b^n = a$. It is written as $\sqrt[n]{a} = b$. When $n=2$, it's the square root; when $n=3$, it's the cube root.

Roots are the inverse operation of exponentiation. For example, $\sqrt{16} = 4$ because $4^2 = 16$, and $\sqrt[3]{27} = 3$ because $3^3 = 27$.

Properties of radicals - Product rule: $\sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b}$ - Quotient rule: $\sqrt[n]{a / b} = \sqrt[n]{a} / \sqrt[n]{b}$ - Fractional exponents: $\sqrt[n]{a} = a^{1/n}$

Newton-Raphson method This calculator uses the Newton-Raphson iterative method to find roots to high precision (tolerance $10^{-12}$). The iteration formula for $\sqrt[n]{a}$ is:

$$x_{k+1} = \frac{(n-1)x_k + a / x_k^{n-1}}{n}$$

Each iteration is shown in the step-by-step display.

Surds A surd is a radical expression that cannot be fully simplified, such as $\sqrt{2}$ or $\sqrt{3}$. These are irrational numbers with non-terminating, non-repeating decimal expansions.

How to use this calculator Select the Square Root or Nth Root tab. Enter the radicand (and degree for nth root). The result appears instantly with Newton-Raphson iteration steps.

Privacy & Performance All calculations run locally in your browser. No data is transmitted, logged, or stored on any server.

🔒 Client-side only⚡ Web Worker processing🌐 Works offline🛡️ Zero server logs

Frequently Asked Questions

Got questions about roots and radicals? We've got answers.

What is a square root?

A square root of a number a is a value b such that b² = a. For example, the square root of 16 is 4 because 4² = 16. Every non-negative number has two square roots (positive and negative), but the principal square root is the non-negative one.

What is the difference between a square root and an nth root?

A square root finds a value that, when multiplied by itself twice (squared), equals the radicand. An nth root finds a value that, when multiplied by itself n times, equals the radicand. For example, the cube root (n=3) of 27 is 3 because 3³ = 27.

Can I take the square root of a negative number?

In real numbers, no — the square root of a negative number is not a real number. However, in complex numbers, it is defined using the imaginary unit i, where √(-1) = i. This calculator works with real numbers only.

What is the Newton-Raphson method?

The Newton-Raphson method is an iterative algorithm for finding successively better approximations to the roots of a real-valued function. For nth roots, it starts with an initial guess and refines it using the formula: x_{k+1} = ((n-1)x_k + a / x_k^(n-1)) / n until the result converges.

What is a surd?

A surd is an expression containing a radical that cannot be simplified to remove the radical. For example, √2 is a surd because it is irrational. Surds are often left in radical form in mathematics rather than being expressed as decimals.

Is this calculator private?

Yes. All calculations run locally in your browser. No data is transmitted or stored on any server.