Simplifying the Root

Part of the Triangles section of Coddy's Geometry journey. Lesson 33 of 71.

Legs 3 and 6 give c = √(9 + 36) = √(45). The number 45 is not a perfect square, but it has a perfect square as a factor: 45 = 9 × 5. The root of a product is the product of the roots, so

√(45) = √(9 × 5) = √(9) × √(5) = 3√(5)

3√(5) means 3 times the root of 5. It is the simplified form, because the number left under the root, 5, has no square factor.

To simplify a root, find the largest perfect square that divides the number under the root. Take the square root of that factor and write it outside the root. For example, √(20) = 2√(5), √(50) = 5√(2), and √(72) = 6√(2). If the number has no perfect square factor bigger than 1, such as 13 or 65, the root cannot be simplified.