Simplifying Radicals Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Simplifying Radicals.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
Simplifying a radical means rewriting it so no perfect-square factor remains under the root sign. For example, √50 = √(25·2) = 5√2. The result — called simplified radical form — has the smallest possible number under the radical.
Look inside the radical for perfect squares hiding as factors. \sqrt{72} contains 36 \times 2, and since \sqrt{36} = 6, you can pull the 6 out: \sqrt{72} = 6\sqrt{2}. Think of it as freeing numbers that are 'ready' to leave the radical.
Read the full concept explanation →How to Use These Examples
- Read the first worked example with the solution open so the structure is clear.
- Try the practice problems before revealing each solution.
- Use the related concepts and background knowledge badges if you feel stuck.
What to Focus On
Core idea: Use the property \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} to separate perfect square factors from the rest.
Common stuck point: Finding the largest perfect square factor. It helps to factor the radicand completely into primes first.
Sense of Study hint: Break the number under the radical into prime factors, then pull out any pair of identical primes.
Worked Examples
Example 1
easySolution
- 1 Step 1: Find the largest perfect square factor: 72 = 36 \times 2.
- 2 Step 2: \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2}.
- 3 Check: 6^2 \times 2 = 72 ✓
Answer
Example 2
mediumExample 3
mediumPractice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
easyExample 2
mediumRelated Concepts
Background Knowledge
These ideas may be useful before you work through the harder examples.