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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.

2√3*a*^{3} + 5*a*√3*a* =

7*a*√3*a*

7*a*√3*a*^{2}

7*a*^{2}√3*a*

7*a*^{2}√3*a*^{2}

2.

√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

√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

Answer (b) is the correct solution

3.

√32 + 9√2 + √18 =

16√2

16√2^{3}

11√2

-16√2^{3}

√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

√32 = √16 • 2 = 4√2

√18 = √9 • 2 = 3√2

4√2 + 9√2 + 3√2 = 16√2

Answer (a) is the correct solution

4.

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

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

Answer (a) 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

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

Answer (b) is the correct solution

6.

2√24*y* - 2√54*y* =

10√6*y*

-2√6*y*

2√6*y*

-10√6*y*

2√24*y* - 2√54*y*

2√24*y* = 2√4 • 6 • *y* = 2 • 2√6*y* = 4√6*y*

2√54*y* = 2√9 • 6 • *y* = 2 • 3√6*y* = 6√6*y*

4√6*y* - 6√6*y* = -2√6*y*

__Solution__: -2√6*y*

Answer (b) is the correct solution

2√24

2√54

4√6

Answer (b) is the correct solution

7.

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

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

Answer (d) is the correct solution

8.

-√6 *x* - √54*x* =

-3√6*x*

2√6*x*

-4√6*x*

4√6*x*

-√6*x* - √54*x*

√54*x* = √9 • 6 • *x* = 3√6*x*

-√6*x* - 3√6*x* = -4√6*x*

Solution: -4√6*x*

Answer (c) is the correct solution

√54

-√6

Solution: -4√6

Answer (c) is the correct solution

9.

2√5*x* + 5√5*x* - 2√5*x* =

7√5*x*

-2√5*x*

5√5*x*

5√5*x*^{3}

2√5*x* + 5√5*x* - 2√5*x*

2√5*x* - 2√5*x* = 0

5√5*x* + 0 = 5√5*x*

__Solution__: 5√5*x*

Answer (c) is the correct solution

2√5

5√5

Answer (c) is the correct solution

10.

√5*a*^{2} + *a*√45 =

3*a*√5^{2}

4*a*√5^{2}

3*a*√5

4*a*√5

√5*a*^{2} + *a*√45

√5 (*a* • *a*) = *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 = 3*a*√5 We now have:

*a*√5 + 3*a*√5 = 4*a*√5

__Solution__: 4*a*√5

Answer (d) is the correct solution

√5 (

Answer (d) is the correct solution

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a^{3}+ 5a√3a2√

a^{2}• 3a+ 5a√3a2>

a√3a+ 5a√3a= 7a√3a7Solution:a√3aAnswer (a) is the correct solution