Addition and Subtraction of Radicals

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. 5 + 3 5 = 4 5
Example 2: Subtract 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.
Example 4: Add/Subtract
18 + 4 32 = 3 2 + 16 2 = 19 2
3 3
3
2 54 + 3 24 − 8 16
Solution: Again, we must first simplify the radicals before adding/subtracting.
Example 5: Add/Subtract
3 3 3 3 3 3 3 3
2 54 + 3 24 − 8 16 = 6 2 + 6 3 − 16 2 = 6 3 − 10 2
3
24 + 4 96 − 18
x x x y
Solution: In order to add/subtract we must have the same index, radicand, and variables
both inside and outside of the radicals.
3
24 + 4 96 − 18 = 2 6 + 16 6 − 3 2 = 18 6 −
3 2
x x x y x x x x y x x 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.
7.
2 2
5x 99y + 2y 44x
6. 14xy 128x − 17 128x
y
3 3 3 3
23 48x y 10xy 6
+ 8.
3 4 3
3 5 2
2000xy − 4y 54xy
9. ( 6 + 2 )( 4 3 + 2 )
10. ( 2 5 − 3)( 4 5 + 2 3)
11. ( 10 + 2 )( 5 − 3)
12. ( 11 − 7 )( 8 − 5)
13. ( ) 2
7 + 2
14. ( ) 2
5 6 −
3