Square Root Calculator

Calculate square roots, cube roots, and custom nth roots of numbers with automatic radical simplification, perfect power identification, and precision breakdowns.

Root Parameters

Live calculation
Quick Numbers:
Square Root (√x)
12
Radical Expression√144
Classification: Perfect Square
Radicand (x)144
Root Degree (n)2
Inverse (xⁿ)20,736
Power (x²)20,736

Root Verification & Exact Identity

• Mathematical identity: (12)^2 = 144 ≈ 144

• Exponent representation: 144^(1/2) = 12

What Is a Square Root?

A square root of a number x is a number y such that y² = x. The radical symbol √ indicates the principal (non-negative) root. More broadly, an nth root ⁿ√x satisfies yⁿ = x.

1. General Nth Root Exponent Definition
ⁿ√x=x^(1/n)where(ⁿ√x)ⁿ = x
2. Radical Simplification Property
√(a × b)=√a × √b→√(36 × 2) = 6√2
Step-by-Step Calculation Breakdown (Example: √72)
Step 1: Identify Largest Perfect Square Factor
• Factors of 72 include: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
• Largest perfect square factor = 36 (since 6² = 36)
Step 2: Factor and Extract Outside the Radical
• √72 = √(36 × 2) = √36 × √2 = 6√2
Step 3: Evaluate Decimal Value
• 6 × 1.41421356... = 8.485281

First 12 Perfect Squares & Cubes Reference

Integer (n)Square (n²)Cube (n³)Square Root (√n²)
1111
2482
39273
416644
5251255
6362166
7493437
8645128
9817299
10100100010
11121133111
12144172812

Frequently Asked Questions

What is a square root in mathematics?
A square root of a number x is a value y such that y² = x. For example, √144 = 12 because 12 × 12 = 144. In the real number system, every positive number has both a positive (principal) square root and a negative square root (±√x). By convention, the radical symbol √ denotes the non-negative principal square root.
What is the square root of a negative number?
Even roots (such as square roots and 4th roots) of negative numbers are undefined in the real number system because squaring any real number always yields a non-negative result. They are expressed using imaginary numbers (e.g. √−9 = 3i). Odd roots (such as cube roots) of negative numbers are real: ∛−27 = −3.
What is a perfect square?
A perfect square is an integer that is the square of another integer. The first 15 positive perfect squares are 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225. This calculator automatically detects perfect squares and cubes.
How do you simplify a radical?
To simplify a square root, factor the number under the radical into its largest perfect square factor. For instance, √72 = √(36 × 2) = √36 × √2 = 6√2. This calculator automatically computes simplified radical forms.
How do you calculate nth roots?
The nth root of x is equivalent to raising x to the power of 1/n: ⁿ√x = x^(1/n). For instance, ⁴√81 = 81^(1/4) = 3.
What is the inverse operation of a root?
Exponentiation is the inverse of extraction of roots. If y = ⁿ√x, then yⁿ = x. For example, (√25)² = 5² = 25.

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