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Mathematics LibreTexts

8.3: Adding and Subtracting Radical Expressions

  • Anonymous
  • LibreTexts

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Learning Objectives
  • Add and subtract like radicals.
  • Simplify radical expressions involving like radicals.

Adding and Subtracting Radical Expressions

Adding and subtracting radical expressions is similar to adding and subtracting like terms. Radicals are considered to be like radicals, or similar radicals, when they share the same index and radicand. For example, the terms 35 and 45 contain like radicals and can be added using the distributive property as follows:

35+45=(3+4)5=75

Typically, we do not show the step involving the distributive property and simply write

35+45=75

When adding terms with like radicals, add only the coefficients; the radical part remains the same.

Example 8.3.1

Add:

32+22

Solution:

The terms contain like radicals; therefore, add the coefficients.

32+22=52

Answer:

52

Subtraction is performed in a similar manner.

Example 8.3.2

Subtract:

2737

Solution:

Answer:

7

If the radicand and the index are not exactly the same, then the radicals are not similar and we cannot combine them.

Example 8.3.3

Simplify:

Solution:

We cannot simplify any further because 5 and 2 are not like radicals; the radicands are not the same.

Answer:

52

Note

Caution

It is important to point out that 5252. We can verify this by calculating the value of each side with a calculator.

520.8252=31.73

In general, note that na±nbna±b.

Example 8.3.4

Simplify:

Solution:

We cannot simplify any further because 36 and 6 are not like radicals; the indices are not the same.

Answer:

2366

Often we will have to simplify before we can identify the like radicals within the terms.

Example 8.3.5

Subtract:

1248

Solution:

At first glance, the radicals do not appear to be similar. However, after simplifying completely, we will see that we can combine them.

Answer:

23

Example 8.3.6

Simplify:

Solution:

Answer:

53

Exercise 8.3.1

Subtract:

25068

Answer

22

Next, we work with radical expressions involving variables. In this section, assume all radicands containing variable expressions are not negative.

Example 8.3.7

Simplify:

Solution:

We cannot combine any further because the remaining radical expressions do not share the same radicand; they are not like radicals. Note that 32x33x32x3x.

Answer:

32x33x

We will often find the need to subtract a radical expression with multiple terms. If this is the case, remember to apply the distributive property before combining like terms.

Example 8.3.8

Simplify:

Solution:

Answer:

Until we simplify, it is often unclear which terms involving radicals are similar.

Example 8.3.9

Simplify:

532y(354y316)

Solution:

Answer:

232y+232

Example 8.3.10

Simplify:

2a125a2ba280b+420a4b

Solution:

Answer:

14a25b

Exercise 8.3.2

Simplify:

45x3(20x380x)

Answer

35x3225x35x

Note

Tip

Take careful note of the differences between products and sums within a radical.

Products Sums
x2y2=xy3x3y3=xy x2+y2(x=y)3x3+y3 neqx+y
Table 8.3.1

The property nab=nanb says that we can simplify radicals when the operation in the radicand is multiplication. There is no corresponding property for addition.

Key Takeaways

  • Add and subtract terms that contain like radicals just as you do like terms. If the index and radicand are exactly the same, then the radicals are similar and can be combined. This involves adding or subtracting only the coefficients; the radical part remains the same.
  • Simplify each radical completely before combining like terms.
Exercise 8.3.3 adding and subtracting like radicals

Simplify.

  1. 93+53
  2. 126+36
  3. 4575
  4. 310810
  5. 646+26
  6. 5101510210
  7. 1376257+52
  8. 10131215+5131815
  9. 65(4335)
  10. 122(66+2)
  11. (25310)(10+35)
  12. (83+615)(315)
  13. 436335+636
  14. 310+53104310
  15. (739433)(39333)
  16. (835+325)(235+6325)
Answer

1. 143

3. 25

5. 6

7. 872

9. 9543

11. 5410

13. 1036335

15. 63933

Exercise 8.3.4 adding and subtracting like radicals

Simplify. (Assume all radicands containing variable expressions are positive.)

  1. 9x+7x
  2. 8y+4y
  3. 7xy3xy+xy
  4. 10y2x12y2x2y2x
  5. 2ab5a+6ab10a
  6. 3xy+6y4xy7y
  7. 5xy(3xy7xy)
  8. 8ab(2ab4ab)
  9. (32x3x)(2x73x)
  10. (y42y)(y52y)
  11. 53x123x
  12. 23y33y
  13. a53b+4a53ba53b
  14. 84ab+34ab24ab
  15. 62a432a+72a32a
  16. 453a+33a953a+33a
  17. (44xy3xy)(244xy3xy)
  18. (566y5y)(266y+3y)
Answer

1. 16x

3. 5xy

5. 8ab15a

7. 9xy

9. 22x+63x

11. 73x

13. 4a53b

15. 132a532a

17. 44xy

Exercise 8.3.5 adding and subtracting rational expressions

Simplify.

  1. 7512
  2. 2454
  3. 32+278
  4. 20+4845
  5. 2827+6312
  6. 90+244054
  7. 4580+2455
  8. 108+48753
  9. 42(2772)
  10. 35(2050)
  11. 316354
  12. 381324
  13. 3135+34035
  14. 310833234
  15. 227212
  16. 350432
  17. 324321848
  18. 6216224296
  19. 218375298+448
  20. 24512+220108
  21. (2363396)(712254)
  22. (2288+3360)(272740)
  23. 3354+532504316
  24. 431622338433750
Answer

1. 33

3. 22+33

5. 5753

7. 55

9. 10233

11. 32

13. 435

15. 23

17. 23362

19. 82+3

21. 8366

23. 2632

Exercise 8.3.6 adding and subtracting rational expressions

Simplify. (Assume all radicands containing variable expressions are positive.)

  1. 81b+4b
  2. 100a+a
  3. 9a2b36a2b
  4. 50a218a2
  5. 49x9y+x4y
  6. 9x+64y25xy
  7. 78x(316y218x)
  8. 264y(332y81y)
  9. 29m2n5m2n+m2n
  10. 418n2m2n8m+n2m
  11. 4x2y9xy216x2y+y2x
  12. 32x2y2+12x2y18x2y227x2y
  13. (9x2y16y)(49x2y4y)
  14. (72x2y218x2y)(50x2y2+x2y)
  15. 12m4nm75m2n+227m4n
  16. 5n27mn2+212mn4n3mn2
  17. 227a3ba48aba144a3b
  18. 298a4b2a162a2b+a200b
  19. 3125a327a
  20. 31000a2364a2
  21. 2x354x2316x4+532x4
  22. 354x33250x6+x232
  23. 416y2+481y2
  24. 532y45y4
  25. 432a34162a3+542a3
  26. 480a4b+45a4ba45b
  27. 327x3+38x3125x3
  28. 324x3128x381x
  29. 327x4y38xy3+x364xyy3x
  30. 3125xy3+38x3y3216xy3+10x3y
  31. (3162x4y3250x4y2)(32x4y23384x4y)
  32. (532x2y65243x6y2)(5x2y6x5xy2)
Answer

1. 11b

3. 3ab

5. 8x5y

7. 202x12y

9. 22mn

11. 2xy2yx

13. 4xy

15. 33m2n

17. 2a3ab12a2ab

19. 23a

21. 7x32x

23. 5y

25. 442a3

27. 2x+23x

29. 7x3xy3y3x

31. 7x36xy6x32xy2

Exercise 8.3.7 discussion board
  1. Choose values for x and y and use a calculator to show that x+yx+y.
  2. Choose values for x and y and use a calculator to show that x2+y2x+y.
Answer

1. Answers may vary


This page titled 8.3: Adding and Subtracting Radical Expressions is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Anonymous via source content that was edited to the style and standards of the LibreTexts platform.

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