Operations with Radicals

Adding radicals works like combining like terms: only identical radicals combine. 3√5 + 2√5 = 5√5, but 3√5 + 2√7 stays as it is.

Simplify first, because unlike radicals often become like ones. √8 + √18 looks unlike, but simplifies to 2√2 + 3√2 = 5√2.

Multiplying is freer: √a · √b = √(ab) for the same index. Distribute and FOIL exactly as with polynomials.

Rationalising removes a radical from a denominator. For a single term, multiply by that radical over itself. For a binomial like 3 + √2, multiply by its conjugate 3 − √2, which produces a difference of squares and clears the radical.

Radicals of the same index multiply and divide via ⁿ√a · ⁿ√b = ⁿ√(ab) and ⁿ√a / ⁿ√b = ⁿ√(a/b), valid for non-negative radicands (or any radicands when n is odd). Different indices must be converted to rational exponents first.

Addition requires identical index *and* identical radicand, since c₁ⁿ√a + c₂ⁿ√a = (c₁ + c₂)ⁿ√a is the distributive property; there is no rule for unlike radicals. Full simplification before combining is therefore mandatory, not cosmetic.

Rationalising a monomial denominator multiplies by ⁿ√(b^{n−k}) chosen to complete the index. For a binomial containing square roots, the conjugate works because (a + √b)(a − √b) = a² − b is rational.

Modern computation does not require rationalised denominators, but the standard form remains conventional and, more usefully, conjugate multiplication is the same technique used to divide complex numbers and to resolve certain indeterminate limits in calculus.