simplify square roots like a pro

A student stares at √72. Their calculator spits out 8.485281, but the homework says "simplify square roots," not "give a decimal." They need simplest radical form, and that means rewriting √72 as 6√2. Three reliable methods exist so you can produce that exact, simplified radical yourself, step by step.

Every positive number has two square roots: the principal (non‑negative) root and its negative counterpart. The radical symbol √ gives only the principal root. When you see √72, you want the positive simplified radical form, not a decimal approximation.

The quickest way to simplify a square root is to factor out the largest perfect square from the number under the radical. A perfect square is an integer whose square root is also an integer: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on.

Identify the largest perfect square that divides your number under the radical evenly. For √72, the perfect squares less than or equal to 72 are 64, 49, 36, 25, 16, 9, 4, and 1. The largest that divides 72 is 36 (since 72 ÷ 36 = 2). Then rewrite √72 as √(36 × 2). Apply the product property of radicals: √(ab) = √a × √b for a ≥ 0, b ≥ 0. So √72 = √36 × √2 = 6√2.

This method works for any number under the radical. If the largest perfect‑square factor is not obvious, list the perfect squares you know and test each until you find the biggest one that fits.

When the number under the radical is large or you prefer a visual breakdown, use a factor tree to find all prime factors. Then pair identical primes and pull each pair out of the radical as a single factor.

Take √72 again. Write 72 as 2 × 36, then 36 as 2 × 18, then 18 as 2 × 9, then 9 as 3 × 3. The prime factors are 2 × 2 × 2 × 3 × 3. Group the pairs: (2 × 2) gives one 2, and (3 × 3) gives one 3. You have one leftover 2 that has no partner. For each pair, take one factor outside the radical: 2 × 3 = 6. The unpaired 2 stays inside: √2. Result: 6√2.

The factor tree is especially useful when the number under the radical has many factors or when you are simplifying radicals with variables, since variables also pair the same way.

√12

The largest perfect square that divides 12 is 4 (since 12 ÷ 4 = 3). Write √12 = √(4 × 3) = √4 × √3 = 2√3. The factor tree: 12 = 2 × 6 = 2 × 2 × 3. Pair the two 2s; one 2 comes out, the 3 stays in: 2√3.

√72

Largest perfect‑square factor: 36. √72 = √(36 × 2) = 6√2. Factor tree: 72 = 2 × 2 × 2 × 3 × 3. Pairs: (2,2) gives one 2; (3,3) gives one 3; leftover 2 stays inside: 6√2.

√200

The largest perfect square dividing 200 is 100 (200 ÷ 100 = 2). √200 = √(100 × 2) = 10√2. Factor tree: 200 = 2 × 100 = 2 × 2 × 50 = 2 × 2 × 2 × 25 = 2 × 2 × 2 × 5 × 5. Pairs: (2,2) gives one 2; (5,5) gives one 5; leftover 2 stays in: 2 × 5 = 10 outside, √2 inside: 10√2.

All three examples produce simplest radical form because no number under the radical contains a perfect square factor and no radical appears in the denominator. OpenStax Intermediate Algebra 2e, section 8.2, lists these same conditions for simplified radical form.

Variables inside a square root follow the same rules. The exponent of the quantity under the radical determines how many pairs you can extract. For √(x⁷), write x⁷ as x⁶ × x. Pair six x's as (x³ × x³) and pull one x³ out. The leftover x stays inside: x³√x.

For x = -3, √((-3)²) = √9 = 3, which is |-3|. When variables are given with constraints (e.g., x ≥ 0), you can drop the absolute value. Otherwise, write |x|.

Apply the same product property: √(x⁹y⁵) = √(x⁸ × x × y⁴ × y) = √(x⁸) × √(y⁴) × √(xy) = x⁴y²√(xy). Every pair of identical factors comes out; any unpaired factor stays under the radical.

The quotient property of radicals, √(a/b) = √a / √b for a ≥ 0 and b > 0, lets you handle fractions. For √(4/9), write √4 / √9 = 2/3. If the number under the radical has a fraction like √(5/2), apply the quotient property: √5 / √2.

Simplest radical form requires that no radical appears in the denominator. To eliminate √2 from the denominator of 1/√2, multiply numerator and denominator by √2: (1 × √2) / (√2 × √2) = √2 / 2. This process is called rationalising the denominator. For denominators with sums like 1/(3 + √5), multiply by the conjugate (3 − √5) to remove the radical from the denominator.

After simplifying, verify three conditions from OpenStax Intermediate Algebra 2e section 8.2: no number under the radical contains a perfect square factor, no number under the radical contains a fraction, and no radical appears in the denominator of a fraction.

A common failure mode is stopping too early. For √72, an answer of 2√18 is not simplest because 18 contains the perfect square 9. Always reduce until the number inside the radical is not divisible by any perfect square greater than 1. If you have √72 = 6√2, test that 2 is not divisible by 4, 9, or any larger perfect square, it is not, so you are done.