Addition and Subtraction of Radicals Addition and subtraction of radicals has more limitations than multiplication and division. The index and radicand of each radical must be exactly the same. Example 1: Add 5 +3 5 Solution: Since each radical is a square root and the radicand is 5 for each term, all we need to do is add the numbers outside of the radicals. Example 2: Subtract 5 +3 5 = 4 5 7− 3 Solution: Each radical is a square root; however, each term has a different radicand. There is nothing to be done so the answer is 7− 3 . Example 3: Add 18 + 4 32 Solution: At first it looks as though we cannot do the addition because the radicands are different; however, we must first simplify each radical before making this determination. 18 + 4 32 = 3 2 + 16 2 = 19 2 Example 4: Add/Subtract 2 3 54 + 3 3 24 − 8 3 16 Solution: Again, we must first simplify the radicals before adding/subtracting. 2 3 54 + 3 3 24 − 8 3 16 = 6 3 2 + 6 3 3 − 16 3 2 = 6 3 3 − 10 3 2 Example 5: Add/Subtract 24 x3 + 4 x 96 x − 18 y Solution: In order to add/subtract we must have the same index, radicand, and variables both inside and outside of the radicals. 24 x3 + 4 x 96 x − 18 y = 2 x 6 x + 16 x 6 x − 3 2 y = 18 x 6 x − 3 2 y Practice Problems Perform the indicated operation and be sure answers are completely simplified. 1. 14 5 − 3 5 − 10 5 2. 3 2 − 6 3 − 8 3 3. 15 18 − 3 50 + 8 98 4. 5 11 + 3 484 − 10 99 5. 5 x 99 y 2 + 2 y 44 x 2 6. 14 xy 128 x3 − 17 128 x5 y 2 7. 23 3 48 x 3 y 3 + 10 xy 3 6 8. 9. ( 6+ 2 4 3+ 2 )( 11. ( 10 + 2 13. ( 7+ 2 ) )( 5− 3 3 2000 xy 4 − 4 y 3 54 xy ) 10. (2 ) 12. ( 14. (5 2 )( 5− 3 4 5+2 3 11 − 7 6− 3 )( ) 2 8− 5 ) )