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
)
)