Radical Operations Examples in Math
Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Radical Operations.
This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.
Concept Recap
Adding, subtracting, and multiplying expressions that contain radicals. Like terms (same radicand) can be combined for addition and subtraction; for multiplication, use \sqrt{a} \cdot \sqrt{b} = \sqrt{ab}.
Treat simplified radicals like variables: 3\sqrt{5} + 2\sqrt{5} = 5\sqrt{5} works just like 3x + 2x = 5x. You can only combine radicals with the SAME radicand. Multiplication is more flexible since \sqrt{a} \cdot \sqrt{b} = \sqrt{ab} always works.
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: Addition/subtraction requires like radicands (simplify first!). Multiplication combines radicands under one radical.
Common stuck point: Before adding or subtracting, simplify each radical first—terms that look unlike may actually be like terms after simplification.
Sense of Study hint: Simplify each radical into simplest form first, then check if they have the same radicand before combining.
Worked Examples
Example 1
easySolution
- 1 Step 1: Both terms have the same radicand \sqrt{5}, so they are like radicals.
- 2 Step 2: Add coefficients: (3 + 7)\sqrt{5} = 10\sqrt{5}.
- 3 Check: Think of \sqrt{5} as a variable: 3a + 7a = 10a ✓
Answer
Example 2
mediumPractice Problems
Try these problems on your own first, then open the solution to compare your method.
Example 1
mediumExample 2
hardBackground Knowledge
These ideas may be useful before you work through the harder examples.