Square Root Calculator

Find principal square roots (√x), cube roots (∛x), and Nth roots (ⁿ√x) with exact simplified radical form (aⁿ√b) and perfect root verification.

Category: Math

How to Use Square Root Calculator

  1. Enter the radicand in 'Number Input' and specify the integer 'Root Index (N)' (use 2 for Square Root √x, 3 for Cube Root ∛x, or any integer N ≥ 1).
  2. Click Calculate to compute the principal N-th root and extract perfect N-th power factors.
  3. Review the decimal root, exact perfect-power check, simplified radical form (aⁿ√b), and x² reference.

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Frequently Asked Questions

How do I calculate Square Roots, Cube Roots, and Nth Roots?

Enter your number (the radicand) in 'Number Input' and set 'Root Index (N)' to 2 for a Square Root (√x), 3 for a Cube Root (∛x), 4 for a Fourth Root (∜x), or any positive integer N ≥ 1 to compute x^(1/N).

How does the calculator simplify radicals into exact form (like √50 = 5√2)?

For integer radicands and root index N, the engine searches for the largest perfect N-th power factor aᴺ that evenly divides your input x = aᴺ × b. It pulls 'a' outside the radical sign and leaves the square-free remainder 'b' inside: ⁿ√x = aⁿ√b (for example, √72 = 6√2 and ∛54 = 3∛2).

Can you take the root of a negative number?

It depends on whether the Root Index N is odd or even: 1) For odd root indices (N = 3, 5, 7, ...), a negative number has a real negative root—for example, the cube root of −27 (∛−27) is −3 because (−3)³ = −27; 2) For even root indices (N = 2, 4, 6, ...), the root of a negative number is not a real number (it is an imaginary number in the complex plane, such as √−144 = 12i).

How does the calculator avoid floating-point errors on perfect cube roots like ∛64?

In standard IEEE-754 floating-point math, Math.pow(64, 1/3) evaluates to 3.9999999999999996 instead of 4. EverySolve checks whether the nearest integer raised to the N-th power equals the integer radicand (4³ = 64) and snaps exact perfect powers to their true integer root.

What is the difference between the principal square root and solving x² = a?

The radical symbol √a denotes the principal (non-negative) root—so √144 is strictly +12. By contrast, the quadratic equation x² = 144 has two solutions: x = +12 and x = −12.