How to Simplify Radicals
Simplifying a radical means rewriting it so the radicand (the number under the radical sign) has no perfect square factors (for square roots) or perfect nth power factors (for nth roots). The process involves factoring the radicand and extracting any perfect powers.
Step-by-Step Method
- Find the prime factorization of the radicand.
- Group factors into sets of n (where n is the index of the radical).
- For each complete group, bring one factor outside the radical.
- Multiply all extracted factors for the coefficient.
- Multiply remaining factors inside the radical for the simplified radicand.
Common Examples
√50
50 = 25 x 2 = 5² x 2
√50 = 5√2
√200
200 = 100 x 2 = 10² x 2
√200 = 10√2
√[3]{54}
54 = 27 x 2 = 3³ x 2
√[3]{54} = 3√[3]{2}
Perfect Squares to Remember
Knowing your perfect squares makes simplification faster: 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225.
Tips for Simplifying
- Always start by finding the prime factorization.
- For square roots, look for pairs of identical prime factors.
- For cube roots, look for triplets of identical prime factors.
- A radical is fully simplified when no perfect nth power divides the radicand.