Lesson 54: Operations with Radicals (Add, Subtract, Multiply, Divide)
Once radicals are simplified, we can perform operations on them: addition, subtraction, multiplication, and division. Similar to algebraic expressions, like radicals can be combined, while multiplication and division follow special rules.
Addition and Subtraction of Radicals
Only radicals with the same radicand and index can be combined:
- √2 + 3√2 = 4√2
- 5√3 − 2√3 = 3√3
- √5 + √2 cannot be simplified further
Multiplication of Radicals
Multiply the coefficients and radicands separately:
- √2 × √3 = √6
- 3√2 × 4√5 = 12√10
- (√2)(√8) = √16 = 4
Division of Radicals
Divide coefficients and radicands separately. Rationalize denominators if needed:
- √18 / √2 = √9 = 3
- (6√3)/(2√2) = 3√6 / 2
- 1 / √2 = √2 / 2 (rationalized)
Examples
Example 1: 2√5 + 3√5 = 5√5
Example 2: √3 × √12 = √36 = 6
Example 3: 4√10 ÷ 2√5 = 2√2
Practice Problems
- √8 + 2√8
- 3√7 − √7
- √5 × √20
- 6√6 ÷ 3√2
- 1 ÷ √3 (rationalize the denominator)