Simplify Radicals
Reduce a square root to simplest radical form a√b by pulling out the largest perfect square.
≈ 8.4853
To simplify a square root, pull out the largest perfect-square factor. For √72, the largest perfect square dividing 72 is 36, so √72 = √(36 × 2) = √36 × √2 = 6√2. The exact simplest radical form is 6√2, which equals ≈ 8.49 as a decimal.
What “simplest radical form” means
A square root is in simplest radical form when the number under the radical (the radicand) has no perfect-square factor other than 1. Simplifying never changes the value — 6√2 and √72 are exactly equal — it just rewrites the root so the inside is as small as possible. This makes radicals easier to compare, add, and use in exact answers.
pull each perfect-square factor a² out of the radical as a
Worked example
Simplify √72:
- 1 Find the largest perfect-square factor. List the perfect squares (4, 9, 16, 25, 36, …) that divide 72. The largest is 36, since 72 = 36 × 2.
- 2 Split the radical into two roots. √72 = √(36 × 2) = √36 × √2, using the product rule √(a·b) = √a × √b.
- 3 Take the perfect square outside. √36 = 6, so √72 = 6√2. The radicand 2 is prime, so it cannot be reduced further — this is simplest radical form. As a decimal, 6√2 ≈ 8.49.
Common radicals in simplest form
Each radicand is factored into a perfect square times a remainder.
| Square root | Perfect-square factor | Simplest form |
|---|---|---|
| √8 | 8 = 4 × 2 | 2√2 |
| √12 | 12 = 4 × 3 | 2√3 |
| √32 | 32 = 16 × 2 | 4√2 |
| √50 | 50 = 25 × 2 | 5√2 |
| √72 | 72 = 36 × 2 | 6√2 |
Perfect squares, already-simplified roots, and denominators
Perfect squares simplify to a plain integer: √49 = 7, because 49 = 7² with nothing left under the radical. Some roots are already simplest — √15 stays √15 because 15 = 3 × 5 has no repeated prime factor, so no perfect square can be pulled out.
Rationalizing denominators is a related cleanup: a fraction like 1 ÷ √2 is usually rewritten by multiplying top and bottom by √2 to give √2 ÷ 2, moving the radical out of the denominator. The simplified value is the same, but a rational denominator is the standard final form.