A root undoes a power
The square root of a number answers “what, squared, gives this?” Since 5^2 = 25, sqrt(25) = 5. The cube root asks the same about cubing: 2^3 = 8 so the cube root of 8 is 2. The general nth root undoes an nth power. In symbols, the nth root of b is the number whose nth power is b.
The whole symbol is a radical. The number under the bar is the radicand, and the small number tucked into the crook — the index — says which root you want. The square root is the default: when no index is written, it’s 2. So in the cube root of 8, the radicand is 8 and the index is 3.
Perfect and not-so-perfect
Some radicands come out clean: sqrt(36) = 6, sqrt(100) = 10. These are perfect squares. Most don’t — sqrt(2), sqrt(7), sqrt(50) — and those are irrational numbers, decimals that run forever without repeating. We don’t round them; we keep them exact in radical form and instead try to make them simpler.
Simplifying with the product and quotient rules
Two rules let you split a radical: sqrt(a·b) = sqrt(a)·sqrt(b) and sqrt(a/b) = sqrt(a)/sqrt(b). To simplify a square root, find the largest perfect-square factor of the radicand, split it off, and take its root out front. The leftover stays under the radical.
sqrt(50) = sqrt(25 · 2) = sqrt(25) · sqrt(2) = 5·sqrt(2) sqrt(72) = sqrt(36 · 2) = 6·sqrt(2) Variables (assume positive): sqrt(x^6) = x^3 (half the even exponent) sqrt(18x^5) = sqrt(9·2·x^4·x) = 3x^2·sqrt(2x) Cube root: cbrt(54) = cbrt(27 · 2) = 3·cbrt(2)