Square Root Calculator – Exact and Decimal Results
Calculate the square root, cube root, or nth root of any number. Shows exact simplification and decimal result.
√ Math📐 √x = x^(1/2) | ∛x = x^(1/3)
Number to find root of
Root degree (2=square, 3=cube)
Please enter valid values.
Formula & Reference
| Variable | Formula | Units |
|---|---|---|
| Square Root Calculator – Exact and Decimal Results | √x = x^(1/2) | ∛x = x^(1/3) | result |
Step-by-Step Examples
Example 1
√144
Square root of 144.
- 144 = 12 × 12
- √144 = 12 — perfect square!
✓ 12
Example 2
√2
Square root of 2.
- √2 = 1.41421356… — irrational number
✓ 1.41421…
Example 3
∛27
Cube root of 27.
- 27 = 3 × 3 × 3 — perfect cube
- ∛27 = 3
✓ 3
Real-World Applications
Students
Calculate square roots and cube roots quickly.
Math Teachers
Demonstrate root concepts with any number.
Programmers
Quick root calculations for algorithms.
Engineers
Calculate dimensions from areas quickly.
Common Mistakes to Avoid
⚠️
Not all square roots are irrational
Perfect squares (4, 9, 16, 25, 36…) have exact integer square roots. Most other numbers have irrational square roots (non-terminating, non-repeating decimals).
⚠️
Negative numbers under square root
In real numbers, √(-1) is undefined. In complex numbers, √(-1) = i (imaginary unit). This calculator works for non-negative real numbers.
Frequently Asked Questions
What is a perfect square? ▾
A number whose square root is an integer. Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144…
How to simplify square roots? ▾
Factor out perfect squares. √72 = √(36×2) = 6√2. √50 = √(25×2) = 5√2.
Is √2 a rational number? ▾
No — √2 is irrational. It cannot be expressed as a fraction. Its decimal expansion is infinite and non-repeating: 1.41421356…
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Formula Explorer connections
Interpretation: This relationship converts, summarizes or checks numerical quantities using standard arithmetic and measurement rules. Assumption: Use consistent units, preserve enough significant digits, and round only the final result unless the method states otherwise.