Mathematics
Simplifying radicals
Simplify a root of any index by pulling factors out, or rationalise a denominator. The prime factorisation is shown beside the answer.
What simplifying a root means
Simplifying a radical means bringing out from under the root sign everything that can come out, leaving as little as possible inside. The operating rule is a single one: factorise the radicand into primes, and every complete group of identical factors, as many as the index, comes out as a single factor. For √72 the factorisation is 2³·3², the 2² and the 3² come out as 2 and 3, and a 2 stays inside: the answer is 6√2.
The simplified form is not a matter of taste. It makes radicals immediately comparable — √72 and 6√2 are the same number, but only the second form adds with other multiples of √2 — and it reduces rounding error, because you work with smaller numbers until the last step.
Rationalising solves a different problem: getting the root out of the denominator. Multiply numerator and denominator by the same root, which is multiplying by one and therefore changes nothing, but turns √c · √c into c. Historically it mattered because dividing by an irrational number by hand is awkward; today it survives because the rationalised form is the canonical one and because it exposes simplifications that would otherwise stay hidden.
Common mistakes
- Distributing the root over a sum: √(a + b) is not √a + √b. The root distributes over products and quotients, never over sums.
- Pulling out a factor that does not form a complete group: from √12 = √(2²·3) only the 2 comes out, not the 3. The 3 appears once and the index is two.
- Forgetting the absolute value when simplifying √(x²): the answer is |x|, not x, because a square root always returns a non-negative value.
Frequently asked questions
How do you simplify a square root?
Factorise the radicand into primes and bring out one factor for each pair of identical factors. For √72 = √(2³·3²) the 2 and 3 come out and a 2 stays in: the answer is 6√2.
How do you rationalise a denominator?
Multiply numerator and denominator by the root in the denominator. From 3/(2√5) you get 3√5/(2·5) = 3√5/10, where the denominator is now a whole number.
Can you take the root of a negative number?
With an odd index yes, and the result is negative: the cube root of −8 is −2. With an even index no, at least not in the reals, because no number raised to an even power gives a negative result.
What is the difference between a root and a fractional exponent?
None: they are two notations for the same operation. The n-th root of a is also written a^(1/n), and that form makes all the properties obvious, because they become the ordinary rules of exponents.
How this calculation works
Simplification: with the radicand factorised as p₁^e₁ · p₂^e₂ · …, each prime comes out of the root raised to the integer part of eᵢ/n and stays inside raised to the remainder of eᵢ/n, where n is the index. Fractional-exponent notation: ⁿ√(a^m) = a^(m/n). Properties: ⁿ√(a·b) = ⁿ√a · ⁿ√b and ⁿ√(a/b) = ⁿ√a / ⁿ√b; there is no matching property for sums. Rationalising a denominator of the form a/(b·√c): multiply top and bottom by √c to get (a·√c)/(b·c), then simplify the fraction. Even-index roots are defined in the reals only for non-negative radicands; odd-index roots are defined for every real radicand, with the sign coming out of the root.