Simplifying Radicals

To simplify a square root, find a perfect square hiding inside.

√72: since 72 = 36 × 2 and 36 is a perfect square, √72 = 6√2.

If you do not spot the largest square immediately, take any square and repeat. 72 = 4 × 18 gives 2√18, then 18 = 9 × 2 gives 2 · 3√2 = 6√2. Same answer.

Radicals add only when the parts under the root match, exactly like like terms: 3√5 + 2√5 = 5√5, but 3√5 + 2√7 stays as it is.

Multiplication is freer: √3 · √12 = √36 = 6.

The product and quotient rules for radicals, √(ab) = √a·√b and √(a/b) = √a/√b, hold for non-negative a and b. They fail for negatives, which is why √(−4)·√(−9) is not √36 — that computation gives 6 instead of the correct −6 and is the standard cautionary example before imaginary numbers are introduced properly.

Simplified form requires: no perfect-square factor under the radical, no fractions under the radical, and no radical in a denominator. The last condition is met by rationalising — multiply by √b/√b for a monomial denominator, or by the conjugate (a − √b) for a binomial one, since (a + √b)(a − √b) = a² − b is rational.

With variables, √(x²) = |x| unless the domain guarantees x ≥ 0. Textbooks often assume non-negative variables to avoid the absolute value, but the assumption is doing real work and should be stated.