ganak.appSquare Root Calculator
Calculate square roots, cube roots, and nth roots of any number. Simplify radicals to exact form (e.g., √72 = 6√2), view prime factorization, step-by-step solutions, and identify perfect squares. Supports imaginary numbers and a complete reference table.
Supports integers, decimals, and negative numbers.
√(144) = 12
Step-by-Step Solution
How the square root is calculated
Formula
√x = x^(1/2)
Step 1: Find √(144)
Step 2: 12² = 144, so √(144) = 12
Verification: 12² = 144 ✓
Perfect Squares Table
Quick reference for perfect square roots
What Is a Square Root?
Understanding radicals and roots
A square root of a number x is a value y such that y × y = x. For example, the square root of 144 is 12 because 12 × 12 = 144. Every positive number has two square roots: one positive (the principal root) and one negative. When we write √x, we mean the principal (positive) root.
Square Root Formula
√x = x^(1/2) = y, where y² = x
More generally, the nth root of x is a value y such that yⁿ = x. Cube roots (∛x) find the value that multiplied by itself three times equals x. This calculator handles any root degree from square roots to arbitrary nth roots.
General nth Root Formula
ⁿ√x = x^(1/n) = y, where yⁿ = x
How to Simplify Square Roots
The prime factorization method
Simplifying a square root means writing it in the form a√b, where b has no perfect square factors. The process uses prime factorization to extract pairs of factors from under the radical.
Worked Example: √72
- Prime factorize: 72 = 2³ × 3²
- Find pairs: 2² gives one pair of 2s, 3² gives one pair of 3s
- Extract pairs: 2 × 3 = 6 comes outside the radical
- Remaining factor: one 2 stays inside → √2
- Result: √72 = 6√2 ≈ 8.485
Worked Example: ∛72
- Prime factorize: 72 = 2³ × 3²
- Find triplets (groups of 3): 2³ gives one group of three 2s
- Extract: 2 comes outside the cube root
- Remaining factors: 3² = 9 stays inside → ∛9
- Result: ∛72 = 2∛9 ≈ 4.160
General rule for nth roots: In the prime factorization, any prime with exponent ≥ n can have groups of n extracted. Each group contributes one copy of that prime outside the radical. Any remaining exponent stays inside.
Types of Numbers and Roots
Perfect, irrational, and imaginary roots
Perfect Roots
A number whose nth root is an integer. For square roots: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 are perfect squares. √49 = 7 exactly.
Irrational Roots
Most square roots are irrational — they cannot be expressed as a fraction and their decimal expansion never terminates or repeats. Examples: √2 ≈ 1.41421, √3 ≈ 1.73205.
Imaginary Roots
The square root of a negative number is imaginary. √(−1) = i, where i is the imaginary unit. For example, √(−25) = 5i. Even roots of negatives always produce imaginary results.
Odd Roots of Negatives
Odd roots (cube root, 5th root, etc.) of negative numbers are real and negative. For example, ∛(−27) = −3, because (−3)³ = −27. This is unique to odd-degree roots.
Common Mistakes to Avoid
Frequent errors when working with square roots
√(a + b) ≠ √a + √b
Example: √(9 + 16) = √25 = 5, but √9 + √16 = 3 + 4 = 7. The square root of a sum is NOT the sum of square roots.
√(a × b) = √a × √b (only for non-negatives)
This property works for positive numbers: √(4 × 9) = √36 = 6 = 2 × 3 = √4 × √9. But be careful with negative numbers — √(−1 × −1) ≠ √(−1) × √(−1).
√(x²) ≠ x (it equals |x|)
The principal square root is always non-negative: √((-5)²) = √25 = 5, not −5. In general, √(x²) = |x| (the absolute value of x).
Confusing ²√ with ½×
√16 = 4 (the square root), not 16/2 = 8 (half the number). Square root and dividing by 2 are completely different operations.
Frequently Asked Questions
Common questions about square roots, radicals, and nth roots