What is a square root?

A square root of a number is a value that, when multiplied by itself, gives the original number. For example, √9 = 3 because 3^2 = 9. Negative numbers do not have real square roots.

A cube root of a number is a value that, when cubed, gives the original number. For example, cbrt(27) = 3 because 3^3 = 27. Negative numbers have real cube roots, e.g. cbrt(-8) = -2.

How to calculate nth roots?

The nth root is a generalization of square and cube roots: nthrt(a) = a^(1/n). When n is even, a must be non-negative; when n is odd, a can be negative.

√ Root Calculator

Calculate square root, cube root, and nth root

Value

Root Type

√64

Square Root

Common Root Values

Expression	Result
√2	1.414214
√3	1.732051
√5	2.236068
cbrt(2)	1.259921
cbrt(10)	2.154435
√10	3.162278

Root Calculator Guide

This calculator computes square roots, cube roots, and nth roots. Enter the radicand and root index to calculate the result.

Root extraction is the inverse of exponentiation, frequently used in geometry, physics formulas, and engineering design. Square and cube roots are the most commonly used.

Root Operations Explained

Root formulas:

- Square root: √x = x^(1/2) - Cube root: cbrt(x) = x^(1/3) - nth root: nthrt(x) = x^(1/n)

Common square roots: - √2 ≈ 1.414 - √3 ≈ 1.732 - √5 ≈ 2.236 - √10 ≈ 3.162

Properties: - √(a×b) = √a × √b - √(a÷b) = √a ÷ √b - Even roots of negative numbers are imaginary - Odd roots of negative numbers are negative

Applications: Pythagorean theorem, standard deviation, circuit impedance calculations.

This calculator is for reference only. Please verify results independently.

FAQ

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Root Calculator Guide

The Root Calculator supports three types of root operations: square root (√), cube root (cbrt), and arbitrary nth root (nthrt). Simply enter a value and select the root type to get a precise result. For nth roots, you can also customize the root index n.

Root operations are the inverse of power operations: if a^n = b, then nthrt(b) = a. Square root is the most common form, found everywhere in geometric calculations (Pythagorean theorem), physics formulas (energy-velocity relationship), and statistical analysis (standard deviation). This tool also provides a quick reference table of common irrational numbers.

Root Operation Formulas

【Square Root】√x = x^(1/2), requires x ≥ 0
【Cube Root】cbrt(x) = x^(1/3), x can be any real number
【Nth Root】nthrt(x) = x^(1/n)
  - When n is even, requires x ≥ 0
  - When n is odd, x can be negative

【Important Relations】(nthrt(x))^n = x
nthrt(x^n) = |x| (take absolute value when n is even)

【Common Irrational Numbers】
√2 ≈ 1.41421356, √3 ≈ 1.73205081
√5 ≈ 2.23606798, cbrt(2) ≈ 1.25992105

Practical Examples

📋

【Example 1】Calculate √64 Input: 64, select square root Result: √64 = 8 Verification: 8^2 = 64 ✓ 【Example 2】Calculate cbrt(-27) Input: -27, select cube root Result: cbrt(-27) = -3 Verification: (-3)^3 = -27 ✓ (Odd roots allow negative results) 【Example 3】Calculate 5th root of 32 Input: 32, select nth root, n=5 Result: 5th root of 32 = 2 Verification: 2^5 = 32 ✓ 【Example 4】Application in Pythagorean theorem Legs a=3, b=4 → Hypotenuse c=√(3^2+4^2)=√25=5

Important Notes

💡

- The radicand of a square root cannot be negative (in real numbers), otherwise the result is imaginary i - Cube roots can be taken of negative numbers: cbrt(-8) = -2 - Even-order roots require a non-negative radicand; odd-order roots have no such restriction - The nth root of 1 is always 1, and the nth root of 0 is always 0 - Root and power are inverse operations, but note domain restrictions

Application Scenarios

- Geometry: Calculate triangle sides using the Pythagorean theorem, circle radii, etc.
- Physics: Derive velocity from kinetic energy (v=sqrt(2E/m)), AC RMS voltage calculation
- Statistics: Standard deviation σ=√(variance), standard error calculation
- Finance: Volatility (annualized standard deviation of returns), option pricing models
- Computer graphics: Vector length |v|=sqrt(x^2+y^2+z^2), normalization

History and Extensions of Root Operations

The history of root operations dates back to ancient Babylon (around 1800 BC), where clay tablets contained approximations of √2. The ancient Greeks proved that √2 is irrational (cannot be expressed as a ratio of two integers), a discovery that triggered the first crisis in mathematics.

In modern mathematics, root operations generalize to fractional exponents: x^(m/n) = nthrt(x^m). This unifies roots within the power operation framework. In the complex number domain, every non-zero complex number has exactly n distinct nth roots, evenly distributed on a circle in the complex plane — this is the concept of roots of unity, with important applications in signal processing (DFT) and number theory.

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