Root Calculator
Find square roots, cube roots and any nth root, with exact simplified surd form where one exists.
How to use this calculator
- 1Enter the number and which root you want — 2 for square root, 3 for cube root.
- 2Where an exact simplified form exists, it is shown alongside the decimal.
- 3The check row confirms the result raised back to that power returns your input.
How the calculation works
ⁿ√x = x^(1/n)- x
- The number under the radical
- n
- The degree of the root — 2 for square, 3 for cube
Roots and exponents are the same operation. Taking the nth root is raising to the power 1/n.
Simplifying a surd means pulling out any prime whose exponent reaches the degree: √72 = √(2³ × 3²) = 6√2, because one 3 and one 2 escape the radical.
Even roots of negative numbers are complex and are rejected here. Odd roots of negative numbers are perfectly real.
Worked example
The square root of 72
- 1.72 factors into 2³ × 3².
- 2.For a square root, every pair of identical primes escapes the radical: the 3² gives a 3, and two of the three 2s give another 2.
- 3.That leaves 6 outside and one 2 inside, so √72 = 6√2.
- 4.Numerically that is 6 × 1.41421… = 8.48528…
Result: 6√2 ≈ 8.4853
What a root undoes
A root reverses exponentiation the way subtraction reverses addition: the square root of a number is whatever value, multiplied by itself, gets back to the original. Roots and exponents are close enough to being the same operation that a root is often written as a fractional exponent instead of a radical sign — the square root of x is just x^(1/2).
Roots outside the math classroom
Square roots in particular turn up wherever a straight-line distance is derived from squared quantities.
- The Pythagorean theorem — finding the length of a diagonal or a hypotenuse from two perpendicular sides always involves a square root.
- Statistics — standard deviation is defined as the square root of variance, specifically so the result comes back in the same units as the original data.
- Physics — root-mean-square values describe quantities like AC voltage and molecular speed, both defined through a square-root step.
- Finance — a compound annual growth rate is essentially an nth root, undoing several years of compounding back down to a single yearly rate.
Why some roots never end, and why that is fine
Most whole numbers are not perfect squares or perfect cubes, so their roots are irrational — decimals that continue forever without repeating. The discovery that the square root of 2 cannot be written as a simple fraction is a famous early result in mathematics, traditionally associated with the Pythagorean school, which had built much of its worldview on the idea that every quantity could be expressed as a ratio of whole numbers.
The decimal shown for an irrational root here is always a very close approximation, never the exact value — the exact value, where one exists in a clean form, is the simplified surd shown alongside it.
What this assumes, and where it stops
Assumptions
- Real-number arithmetic. Complex results are rejected rather than approximated.
Limitations
- Even roots of negative numbers are complex and are not computed.
- Surd simplification is attempted only for whole numbers below 10¹².
- Irrational roots are shown as decimal approximations, accurate to about 15 significant figures.
Common questions
Why does a square root have two answers?
Because both 8 and −8 square to 64. By convention the radical symbol √ denotes only the non-negative root, so √64 = 8. When solving an equation like x² = 64, however, you must consider both x = 8 and x = −8.
How do I simplify a square root by hand?
Factor the number into primes, then pull out any prime appearing twice. For √72: 72 = 2 × 2 × 2 × 3 × 3, so a pair of 2s and a pair of 3s escape as 2 × 3 = 6, leaving one 2 behind. The answer is 6√2.
Formula and content last reviewed on .
Results are estimates for information only, not professional advice.
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