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In order to add radicals, there ** must** be “like” terms in the problem. A problem with like terms would be 5

Let’s look at the next square root problem of 6√2 + 8√2. The like term here is √2. In addition to the √2 being like terms, so are the coefficient numbers “6” and “8” so you get to add those together, i.e., 6 + 8 = 14. The 14 is then placed before √2 and the solution is **14√2.**

The same rule holds true for subtraction.

There ** must** be “like” terms in the problem. Knowing this, if we look at the problem of: 9√

Now let’s look at another problem. 10√12 + 2√3. __You cannot add these binomials because the terms are not like terms__. However, you can simplify 10√12 to read 10√4 ● 3 and then the number 4 is the perfect square root of 2 so the 2 moves to the left of the √ symbol so that the problem now looks like: 10 ● 2√3. Next, 10 ● 2 = 20 giving you 20√3. Now we go back to the original problem and rewrite is as **20√3 + 2√3.** We have √3 as our like term and then 20 + 2 = 22 which is placed before our like term giving us the solution of **22√3.**

For each radical expression shown in the questions below, add or subtract to find the solution.

1.

28√*x* + 31√*x*

59√*x*

59√*x*^{2}

59√*x* + *x*

59

2.

36√9 - 17√9 =

19√18

19√9

19√81

36-17√9

36√9 - 17√9 has like terms, i.e., √9 so the coefficient numbers 36 and 17 can be subtracted or 36 - 17 = 19 and the like term is added.

__Solution__: 19√9

Answer (b) is the correct solution

Answer (b) is the correct solution

3.

√144 - √64

√-80

-√80

4√

8√12

√144 - √64

As we do not have like terms we must simplify as follows:

√144 has a square root as 12 ● 12 = 144. Therefore, the square 12 is then placed before or to the left of the √ symbol giving us 12√

Next, √64 also has a square root as 8 ● 8 = 64. The square 8 is then placed before or to the left of the √ symbol giving us 8√

We now have like terms of √ and the problem can be rewritten as:

12√ - 8√ = 4√

Solution: 4√

Answer (c) is the correct solution

As we do not have like terms we must simplify as follows:

√144 has a square root as 12 ● 12 = 144. Therefore, the square 12 is then placed before or to the left of the √ symbol giving us 12√

Next, √64 also has a square root as 8 ● 8 = 64. The square 8 is then placed before or to the left of the √ symbol giving us 8√

We now have like terms of √ and the problem can be rewritten as:

12√ - 8√ = 4√

Solution: 4√

Answer (c) is the correct solution

4.

5√3 + 9√3 =

14√ 3 + 3

5 + 9√32

14√6

14√3

5√3 + 9√3 has like terms, i.e., √3 so the coefficient numbers 5 and 9 can be added or 5 + 9 = 14 and the like term is added.

__Solution__: 14√3

Answer (d) is the correct solution

Answer (d) is the correct solution

5.

199√*xy* - 99√*xy*

100√*xy*

√*xy*

√100*xy*

100√*xy*^{2}

199√*xy* - 99√*xy* has like terms, i.e., √*xy* so the coefficient numbers 199 and 99 can be subtracted or 199 - 99 = 100 and the like term is added.

__Solution__: 100√*xy*

Answer (a) is the correct solution

Answer (a) is the correct solution

6.

6√125 - 4√5

2√5

26√25

26√5

-24√5

6√125 - 4√5

As we do not have like terms we must simplify as follows:

6√125 = 6√25 ● 5

As 25 is a perfect square in that 5 ● 5 = 25, the 5 goes before or to the left of the √ symbol giving you 6 ● 5√5

6 ● 5 = 30 and then add on the √5 giving you 30√5. Now the problem can be rewritten as: 30√5 - 4√5 = 26√5

__Solution__: 26√5

Answer (c) is the correct solution

As we do not have like terms we must simplify as follows:

6√125 = 6√25 ● 5

As 25 is a perfect square in that 5 ● 5 = 25, the 5 goes before or to the left of the √ symbol giving you 6 ● 5√5

6 ● 5 = 30 and then add on the √5 giving you 30√5. Now the problem can be rewritten as: 30√5 - 4√5 = 26√5

Answer (c) is the correct solution

7.

12√12 + 23√12 =

35√12 + 12

35√12

12 + 23√12 + 12

12 + 23√12

12√12 + 23√12 has like terms, i.e., √12 so the coefficient numbers 12 and 23 can be added or 12 + 23 = 35 and the like term is added.

__Solution__: 35√12

Answer (b) is the correct solution

Answer (b) is the correct solution

8.

45√72 - 221√2

-178√70

-178√72 - 2

49√2

49√6^{2}

45√72 - 221√2

As we do not have like terms we must simplify as follows:

45√72 = 45√36 ● 2

As 36 is a perfect square in that 6 ● 6 = 36, the 6 goes before or to the left of the √ symbol giving you 6 ● 45√2

6 ● 45 = 270 and then add on the √2 giving you 270√2. Now the problem can be rewritten as: 270√2 - 221√2 = 49√2

__Solution__: 49√2

Answer (c) is the correct solution

As we do not have like terms we must simplify as follows:

45√72 = 45√36 ● 2

As 36 is a perfect square in that 6 ● 6 = 36, the 6 goes before or to the left of the √ symbol giving you 6 ● 45√2

6 ● 45 = 270 and then add on the √2 giving you 270√2. Now the problem can be rewritten as: 270√2 - 221√2 = 49√2

Answer (c) is the correct solution

9.

5*a*√*pq* + 7*a*√*pq*

12*a*^{2}√*pq*

12*a*√*pq*^{2}

12*a*^{2}√*pq*^{2}

12*a*√*pq*

5*a*√*pq* + 7*a*√*pq* has like terms, i.e., *a* and√*pq* so the coefficient numbers 5 and 7 can be added or 5*a* + 7*a* = 12*a* and the like term is added.

__Solution__: 12*a*√*pq*

Answer (d) is the correct solution

Answer (d) is the correct solution

10.

10√48 + 15√3

-5√3

55√3

25√51

40√3

10√48 + 15√3

As we do not have like terms we must simplify as follows:

10√48 = 10√16 ● 3

As 16 is a perfect square in that 4 ● 4 = 16, the 4 goes before or to the left of the √ symbol giving you 10 ● 4√3

10 ● 4 = 40 and then add on the √3 giving you 40√3. Now the problem can be rewritten as: 40√3 + 15√3 = 55√3

__Solution__: 55√3

Answer (b) is the correct solution

As we do not have like terms we must simplify as follows:

10√48 = 10√16 ● 3

As 16 is a perfect square in that 4 ● 4 = 16, the 4 goes before or to the left of the √ symbol giving you 10 ● 4√3

10 ● 4 = 40 and then add on the √3 giving you 40√3. Now the problem can be rewritten as: 40√3 + 15√3 = 55√3

Answer (b) is the correct solution

x+ 31√xhas like terms, i.e., √xso the coefficient numbers 28 and 31 can be added or 28 + 31 = 59 and the like term is added.59√Solution:xAnswer (a) is the correct solution