Simplifying Radicals – Writing Roots in Simplest Form

A radical is in simplest form when its radicand has no perfect square factor left inside it (other than 1). √72 isn't wrong exactly, but it's not simplified either – since 72 = 36 × 2, and 36 is a perfect square, √72 can be rewritten as 6√2, a cleaner form that's easier to work with and to compare against other radicals.

The Simplifying Method

Find the largest perfect square factor of the radicand, split the radical using √(a × b) = √a × √b, and pull the perfect square's root out front.

Simplify √50.

50 = 25 × 2, and 25 is a perfect square. √50 = √25 × √2 = 5√2. Simplified: 5√2.

Checking If a Radical Is Already Simplified

Is √15 already in simplest form?

15 = 3 × 5, and neither factor is a perfect square (other than 1). So √15 is already in simplest form.

Real-Life Application

Algebra and geometry: presenting exact answers (like triangle side lengths) in a standard, comparable form.
Exams: simplified radical form is usually required for full marks.
Combining terms: simplifying first makes it possible to add or compare radicals, covered next.

Key Takeaways

A simplified radical has no perfect square factor left in the radicand.
Splitting off the largest perfect square factor gives the form a√b.
Operations with radicals, covered next, rely on radicals being simplified first.

Practice: Simplifying Radicals

Simplify the Radical