Roots as the Inverse of Powers
A root undoes a power. Squaring 7 gives 49; the square root of 49 undoes that and returns 7. Cubing 4 gives 64; the cube root of 64 returns 4. Every root is defined this way, as the number that, raised to the matching power, returns the original value.
Formulas
∛x = x^(1/3)
ⁿ√x = x^(1/n)
(ⁿ√x)ⁿ = x
Writing a root as a fractional exponent, x^(1/n), is not just notation — it lets every rule of exponents apply to roots directly. Multiplying two square roots is the same as taking the square root of the product, because x^(1/2) × y^(1/2) = (xy)^(1/2), one of the more useful simplifications in algebra.
Simplifying a Square Root
A square root simplifies by pulling out any perfect-square factor, leaving the smallest possible number under the radical sign — the exact value, not a rounded decimal.
√50 = 5√2
50 factors as 25 × 2, and 25 is a perfect square. Pulling its root (5) outside leaves 2 under the radical: √50 = √25 × √2 = 5√2, an exact form rather than the decimal approximation 7.071...
√12 = 2√3
12 factors as 4 × 3, and 4 is a perfect square. The root of 4 (which is 2) comes out, leaving 3 inside: √12 = 2√3.
√17 cannot be simplified
17 is prime, with no perfect-square factor other than 1. Its square root is already in simplest radical form and can only be expressed as a decimal approximation beyond that.
Why Most Square Roots Never Terminate
Only perfect squares — 1, 4, 9, 16, 25, and so on — have a whole-number square root. Every other positive integer has an irrational square root: a decimal that continues forever without repeating, proven mathematically rather than simply unobserved in practice.
The proof that √2 is irrational dates to ancient Greek mathematics and is one of the oldest results in number theory, traditionally credited to the Pythagoreans. It matters practically too: any calculation relying on a square root — the Pythagorean theorem, standard deviation, distance formulas — is working with a value that a calculator can only ever approximate to some fixed number of decimal places, never state exactly as a terminating decimal.