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If you have taken the Simple Adding and Subtracting Radicals quiz, that quiz will be a good foundation for tackling this advanced adding and subtracting of radicals. Here we will mostly be adding and subtracting multiple radicals.

Remember, to add or subtract radicals, there must be “like” terms in the problem.

Let’s now work the following practice problem.

5√8 + 6√25 + 7√98
5√8 = 5√4 • 2 = 5 • 2√2 = 10√2
6√25 = 6 • 5 = 30
7√98 = 7√49 • 2 = 7 • 7√2 = 49√2
10√2 + 49√2 = 59√2
Solution: 59√2 + 30

Note that “30” is not a like number to 59√2 so they are NOT added.

1.
√32 + 9√2 + √18 =
16√2
16√23
11√2
-16√23
√32 + 9√2 + √18
√32 = √16 • 2 = 4√2
√18 = √9 • 2 = 3√2
4√2 + 9√2 + 3√2 = 16√2
Solution: 16√2
Answer (a) is the correct solution
2.
2√5x + 5√5x - 2√5x =
7√5x
-2√5x
5√5x
5√5x3
2√5x + 5√5x - 2√5x
2√5x - 2√5x = 0
5√5x + 0 = 5√5x
Solution: 5√5x
Answer (c) is the correct solution
3.
2√18 - 2√12 + 2√18 =
12√2 - 4√3
8√9 - 6√4
7√2 - 4√3
12√2 + 4√3
2√18 - 2√12 + 2√18
2√18 + 2√18 = 4√18
4√18 - 2√12
4√18 = 4√9 • 2 = 4 • 3√2 = 12√2
2√12 = 2√4 • 3 = 2 • 2√3 = 4√3
12√2 - 4√3
Solution: 12√2 - 4√3
Answer (a) is the correct solution
4.
2√20 + √20 + 3√20 - 2√45 =
-6√5
18√5
-√5
6√5
2√20 + √20 + 3√20 - 2√45
2√20 + √20 + 3√20 = 6√20
6√20 - 2√45
6√20 = 6√4 • 5 = 6 • 2√5 = 12√5
2√45 = 2√9 • 5 = 2 • 3√5 = 6√5
12√5 - 6√5 = 6√5
Solution: 6√5
Answer (d) is the correct solution
5.
3√18 + 3√12 + 3√18 =
6√18 + 6√3
18√2 + 6√3
6√18 + 3√12
18√2 + 12√3
3√18 + 3√12 + 3√18
3√18 + 3√18 = 6√18
6√18 = 6√9 • 2 = 6 • 3√2 = 18√2
3√12 = 3√4 • 3 = 3 • 2√3 = 6√3
18√2 + 6√3
Solution: 18√2 + 6√3
Answer (b) is the correct solution
6.
√5a2 + a√45 =
3a√52
4a√52
3a√5
4a√5
√5a2 + a√45
√5 (aa) = a√5
a√45 = a√9 • 5 (as 9 is a perfect square of 3, the 3 goes to the left of the √ symbol)
a√9 • 5 = 3a√5 We now have:
a√5 + 3a√5 = 4a√5
Solution: 4a√5
Answer (d) is the correct solution
7.
2√24y - 2√54y =
10√6y
-2√6y
2√6y
-10√6y
2√24y - 2√54y
2√24y = 2√4 • 6 • y = 2 • 2√6y = 4√6y
2√54y = 2√9 • 6 • y = 2 • 3√6y = 6√6y
4√6y - 6√6y = -2√6y
Solution: -2√6y
Answer (b) is the correct solution
8.
-√6 x - √54x =
-3√6x
2√6x
-4√6x
4√6x
-√6x - √54x
√54x = √9 • 6 • x = 3√6x
-√6x - 3√6x = -4√6x
Solution: -4√6x
Answer (c) is the correct solution
9.
√900 + √400 + √2500 + √1600 =
10√140
140
√140
10√90 + 40 + 250 + 160
√900 + √400 + √2500 + √1600
√900 = √100 • 9 = 10√9 = 10 • 3 = 30
√400 = √100 • 4 = 10√4 = 10 • 2 = 20
√2500 = √100 • 25 = 10√25 = 10 • 5 = 50
√1600 = √100 • 16 = 10√16 = 10 • 5 = 40
30 + 20 + 50 + 40 = 140
Solution: 140
Answer (b) is the correct solution
10.
2√3a3 + 5a√3a =
7a√3a
7a√3a2
7a2√3a
7a2√3a2
2√3a3 + 5a√3a
2√a2 • 3a + 5a√3a
2>a√3a + 5a√3a = 7a√3a
Solution: 7a√3a
Answer (a) is the correct solution
Author:  Christine G. Broome

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