Roots in simple words
A root undoes a power. The square root of a number is the number that, times itself, gives that number. Because 5 × 5 = 25, the square root of 25 is 5, written √25 = 5.
A cube root uses three copies: 2 × 2 × 2 = 8, so the cube root of 8 is 2, written ³√8 = 2. The small number on the root sign (the index) tells you how many copies to use.
Most numbers do not have a whole-number root. √72 is about 8.485, because 8.485 × 8.485 is very close to 72.
You can often make a root simpler. 72 = 36 × 2, and √36 = 6, so √72 = 6√2. That means 6 times the square root of 2.
Formula and method
The nth root of a satisfies xⁿ = a. The calculator returns the principal nonnegative root for an even index and the signed real root for an odd index. Factorization extracts groups of n equal prime factors from an integer radical.
Worked example
72 = 2³ × 3², so √72 = 6√2 ≈ 8.48528137. The cube root of −125 is −5.
Inputs and limitations
An even root of a negative number is not real. The index must be an integer from 1 to 100. Exact radical simplification is limited to integer radicands with magnitude at most 10¹²; larger or decimal values receive a numerical root.
Simplifying by grouping prime factors
Factor the radicand into primes, then split each exponent into complete groups of n and a leftover. Each complete group leaves the radical as a single factor. For √72: 72 = 2³ × 3² = (2 × 3)² × 2, so √72 = 6√2.
For a cube root, group in threes: 72 = 2³ × 3², so ∛72 = 2·∛9. A coefficient multiplies what comes out: 3√72 = 3 × 6√2 = 18√2.