1. Operations with Radicals
GSE Honors Algebra II
Keeper 19

2. Adding or Subtracting Radical Expressions
**You can only add or subtract radical expressions if they have common radicands and indexes.**

3. Steps for Adding or Subtracting Radical Expressions
Step 1: Simplify all radicals completely. Step 2: Combine the outside terms of common radicands. The radicand does NOT change.

4. Example 1

5. Example 2
3 4 2 −5 2

6. Example 3

7. Example 4
− 4 243

8. Example 5
− −

9. Multiplying Radical Expressions
**You can only multiply radicals with the same index.**

10. Steps for Multiplying Radical Expressions
Step 1: Multiply whatever is “outside” of the radical. Step 2: Multiply whatever is on the “inside” of the radical. Step 3: Simplify the radical completely.

11. Example 6
4 2 ∙3 10

12. Example 7
𝑎 2 𝑏 ∙4 3 5𝑎𝑏 𝑐 2

13. Example 8
− 7

14. Example 9

15. Check for Understanding
3 45 𝑥 7 −2𝑥 20 𝑥 5
−2 45 −3 20 −2 6
−3 6 3 − − 6 320
3 3 ∙ 3 9
4 𝑥 𝑥 4 𝑦 ∙2 3 6 𝑥 2 𝑦 3
2𝑥 18𝑥𝑦 ∙5𝑥 6 𝑥 5 𝑦
−3 5
−6 3 81

16. Dividing Radical Expressions
Step 1: Use the Quotient Property of Radicals to rewrite the expression. This property allows you to write a fraction under one big radical or the numerator and denominator under separate radicals. 2𝑥 5 = 2𝑥 5 Step 2: Simplify the numerator. Step 3: Simplify the denominator. You MUST remember that you can NEVER leave a radical in the denominator!

17. Example 10
2 𝑥 2 9 𝑦 2
What if the radical in the denominator does NOT eliminate when we simplify?

18. You must RATIONALIZE THE DENOMINATOR!
*Think about how many of the number/variable we have in the denominator and how many more you need to match the index. Let's call this "just enough."
Step 1: Multiply the numerator and denominator by "just enough" to eliminate the radical in the denominator.
Step 2: Simplify the radical in the numerator.
Step 3: Simplify any outside terms.

19. Example 11
4 20

20. Example 12
20𝑥 50 𝑥 2

21. Special Cases:
When there is a radical being added or subtracted in the denominator, we must multiply by the RADICAL CONJUGATE in order to rationalize the denominator.
3+ 2 →
4− 5 →

22. Example 13
5 2− 7

23. Example 14
4 −2+3 5

24. Example 15
3 𝑥 2𝑦 2

25. Check for Understanding
3 7
3 𝑏 4 𝑎
5𝑥 3 10𝑥 𝑦 3
2 9− 2
4 𝑥 2 𝑦 𝑥 𝑦 2