Square Root Calculator
Calculate square root (√), cube root (∛) and any nth root instantly with decimal precision, perfect-square check and verification.
Negative numbers are only valid for odd-degree roots (cube root, 5th root...).
Quick Example
This square root calculator finds the square root (√), cube root (∛) and any nth root of a number with up to ten digits of decimal precision. It tells you instantly whether the result is a whole number, shows the exact decimal value and verifies the answer by raising it back to the original power. Beyond simple roots, the explanations below cover the algebra of radical numbers — multiplying, dividing and adding radical expressions — so you can both compute and understand.
What Is a Square Root?
A square root (√) is a value that, when multiplied by itself, gives the original number. √25 = 5 because 5² = 25. It is the inverse operation of squaring. Numbers whose root is a whole number are called perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144… The roots of all other numbers are irrational: √2 ≈ 1.41421356… continues forever without repeating.
A cube root (∛) is the inverse of cubing: ∛27 = 3 because 3³ = 27. More generally, the nth root is written ⁿ√x = x^(1/n), which is exactly what the "Other" tab and the cube root calculator mode compute for any degree from 2 up to 100.
How to Find a Square Root
There are several reliable ways to find a square root by hand, and this tool automates the fastest one:
- Memorise perfect squares: √1=1, √4=2, √9=3, √16=4, √25=5, √36=6, √49=7, √64=8, √81=9, √100=10, √121=11, √144=12.
- Prime factorisation: √72 = √(4×9×2) = 2×3×√2 = 6√2 — pull out perfect-square factors to simplify the radical.
- Estimate by bracketing: place the number between two known perfect squares. √50: 7²=49 < 50 < 64=8² → √50 ≈ 7.07.
- Use a calculator or this tool: get the decimal value to ten digits instantly, with a perfect-square check and verification.
What Are Radical Numbers?
Radical numbers are numbers written under a radical (root) sign, such as √2, √7 or ∛5. The number under the sign is called the radicand and the small index tells you the degree of the root. Working with radical numbers follows a clear set of rules for simplifying, multiplying, dividing and adding, which the sections below summarise. A radical that cannot be simplified to a whole number — like √2 or √3 — represents an irrational value.
Multiplying Square Roots and Radical Expressions
Multiplying square roots uses the product rule √a × √b = √(a×b). Example: √3 × √12 = √36 = 6. The same logic extends to multiplying radical expressions with coefficients: 2√3 × 5√2 = (2×5)√(3×2) = 10√6. Roots of the same degree multiply directly; different-degree roots must first be rewritten in exponent form, a^(1/n).
Dividing Square Roots and Radical Expressions
Dividing square roots uses the quotient rule √a ÷ √b = √(a/b). Example: √50 ÷ √2 = √25 = 5. When dividing radical expressions, divide the coefficients and the radicands separately: 12√10 ÷ 3√2 = (12÷3)√(10÷2) = 4√5. If a radical remains in the denominator, multiply top and bottom by it to rationalise — for instance 1/√2 = √2/2.
Adding Square Roots and Radical Expressions
Adding square roots only works when the radicands match — these are called like radicals: 3√5 + 7√5 = 10√5. Different radicals cannot be combined, so √2 + √3 stays as it is. The same rule governs adding radical expressions: simplify each term first, then combine only the like terms. A very common mistake is assuming √(a+b) = √a + √b — this is never true.
Square Root Table (1–100)
| Number | Square Root | Perfect Square? | Cube Root |
|---|---|---|---|
| 1 | 1.000 | Yes | 1.000 |
| 2 | 1.4142… | No | 1.2599… |
| 4 | 2.000 | Yes | 1.5874… |
| 8 | 2.8284… | No | 2.000 |
| 9 | 3.000 | Yes | 2.0801… |
| 16 | 4.000 | Yes | 2.5198… |
| 25 | 5.000 | Yes | 2.9240… |
| 27 | 5.1961… | No | 3.000 |
| 100 | 10.000 | Yes | 4.6415… |
Are Square Roots Rational?
It depends on the number. The root of a perfect square is rational (√9 = 3). The root of a non-perfect square is irrational (√2, √3, √5, √7) — a non-terminating, non-repeating decimal. This was proven mathematically in ancient Greece. The cube root calculator mode follows the same principle: ∛8 = 2 is rational, while ∛9 ≈ 2.0801… is irrational.
Where Roots Are Used in Real Life
- Geometry / Pythagoras: Hypotenuse c = √(a² + b²). Distance calculations always rely on a radical.
- Statistics: Standard deviation is the root of variance: σ = √(Σ(xᵢ−x̄)²/n).
- Physics & engineering: RMS values, impedance and resonant frequency formulas all use radicals.
- Computer graphics: Vector magnitude, normalisation and distance use the √ operation.
- Finance: The Sharpe ratio and volatility scaling apply a square root to time.
Frequently Asked Questions About the Square Root Calculator
A square root (√) is a value that, when multiplied by itself, gives the original number. √25 = 5 because 5² = 25. Numbers whose square root is whole are perfect squares: 1, 4, 9, 16, 25... Others are irrational.
Methods: 1) Memorise perfect squares (√144 = 12). 2) Prime factorisation (√72 = 6√2). 3) A calculator or this tool gives the decimal value instantly. 4) Long division for manual approximation.
Multiplication rule: √a × √b = √(a×b). Example: √3 × √12 = √36 = 6. Roots of the same degree multiply directly; different-degree roots use exponent form a^(1/n).
Only like radicals can be added: 2√3 + 5√3 = 7√3. Different radicals cannot be combined: √2 + √3 stays as is. Simplify first to find matching terms.
Square roots of perfect squares are rational: √9 = 3. Square roots of non-perfect squares are irrational: √2 ≈ 1.41421356... continues forever without repeating.
In real numbers, even-degree roots of negatives are undefined. Odd-degree roots (cube root, 5th root...) are valid: ∛(-8) = -2. In complex numbers, √(-1) = i.
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