Elementary Algebra 2e · free preview

9.3 Add and Subtract Square Roots

Learning objectives: Add and subtract like square roots; Add and subtract square roots that need simplification

1. Learning objectives

Learning objectives

Learning objectives

  • Add and subtract like square roots
  • Add and subtract square roots that need simplification

We know that we must follow the order of operations to simplify expressions with square roots. The radical is a grouping symbol, so we work inside the radical first. We simplify 2+7\sqrt{2+7} in this way:

2+7Addinsidetheradical.9Simplify.3\begin{array}{cccl} & & & \sqrt{2+7} \\ \mathrm{Add inside the radical.} & & & \sqrt{9} \\ \mathrm{Simplify.} & & & 3 \\ \end{array}

So if we have to add 2+7\sqrt{2}+\sqrt{7}, we must not combine them into one radical.

2+7≠2+7\sqrt{2}+\sqrt{7}\ne\sqrt{2+7}

Trying to add square roots with different radicands is like trying to add unlike terms.

But,justlikewecanaddx+x,wecanadd3+3.x+x=2x3+3=23\begin{array}{llll} \mathrm{But, just like we can add} & x+x, & \mathrm{we can add} & \sqrt{3}+\sqrt{3}. \\ & x+x=2x & & \sqrt{3}+\sqrt{3}=2\sqrt{3} \\ \end{array}

Adding square roots with the same radicand is just like adding like terms. We call square roots with the same radicand like square roots to remind us they work the same as like terms.

Square roots with the same radicand are called like square roots.

We add and subtract like square roots in the same way we add and subtract like terms. We know that 3x+8x3x+8x is 11x11x. Similarly we add 3x+8x3\sqrt{x}+8\sqrt{x} and the result is 11x.11\sqrt{x}.

2. Add and Subtract Like Square Roots

Add and Subtract Like Square Roots

Think about adding like terms with variables as you do the next few examples. When you have like radicands, you just add or subtract the coefficients. When the radicands are not like, you cannot combine the terms.

3. Example 9.29

Example 9.29

Example 9.29

Simplify: 22−722\sqrt{2}-7\sqrt{2}.

Solution

22−722\sqrt{2}-7\sqrt{2}
Since the radicals are like, we subtract the coefficients.−52-5\sqrt{2}

4. Example 9.30

Example 9.30

Example 9.30

Simplify: 3y+4y3\sqrt{y}+4\sqrt{y}.

Solution

3y+4y3\sqrt{y}+4\sqrt{y}
Since the radicals are like, we add the coefficients.7y7\sqrt{y}

5. Example 9.31

Example 9.31

Example 9.31

Simplify: 4x−2y4\sqrt{x}-2\sqrt{y}.

Solution

4x−2y4\sqrt{x}-2\sqrt{y}
Since the radicals are not like, we cannotsubtract them. We leave the expression as is.4x−2y4\sqrt{x}-2\sqrt{y}

6. Example 9.32

Example 9.32

Example 9.32

Simplify: 513+413+2135\sqrt{13}+4\sqrt{13}+2\sqrt{13}.

Solution

513+413+2135\sqrt{13}+4\sqrt{13}+2\sqrt{13}
Since the radicals are like, we add the coefficients.111311\sqrt{13}

7. Example 9.33

Example 9.33

Example 9.33

Simplify: 26−66+332\sqrt{6}-6\sqrt{6}+3\sqrt{3}.

Solution

26−66+332\sqrt{6}-6\sqrt{6}+3\sqrt{3}
Since the first two radicals are like, wesubtract their coefficients.−46+33-4\sqrt{6}+3\sqrt{3}

8. Example 9.34

Example 9.34

Example 9.34

Simplify: 25n−65n+45n2\sqrt{5n}-6\sqrt{5n}+4\sqrt{5n}.

Solution

25n−65n+45n2\sqrt{5n}-6\sqrt{5n}+4\sqrt{5n}
Since the radicals are like, we combine them.05n0\sqrt{5n}
Simplify.0

When radicals contain more than one variable, as long as all the variables and their exponents are identical, the radicals are like.

9. Example 9.35

Example 9.35

Example 9.35

Simplify: 3xy+53xy−43xy\sqrt{3xy}+5\sqrt{3xy}-4\sqrt{3xy}.

Solution

3xy+53xy−43xy\sqrt{3xy}+5\sqrt{3xy}-4\sqrt{3xy}
Since the radicals are like, we combine them.23xy2\sqrt{3xy}

10. Add and Subtract Square Roots that Need Simplification

Add and Subtract Square Roots that Need Simplification

Remember that we always simplify square roots by removing the largest perfect-square factor. Sometimes when we have to add or subtract square roots that do not appear to have like radicals, we find like radicals after simplifying the square roots.

11. Example 9.36

Example 9.36

Example 9.36

Simplify: 20+35\sqrt{20}+3\sqrt{5}.

Solution

20+35\sqrt{20}+3\sqrt{5}
Simplify the radicals, when possible.4⋅5+35\sqrt{4}\cdot\sqrt{5}+3\sqrt{5}
25+352\sqrt{5}+3\sqrt{5}
Combine the like radicals.555\sqrt{5}

12. Example 9.37

Example 9.37

Example 9.37

Simplify: 48−75\sqrt{48}-\sqrt{75}.

Solution

48−75\sqrt{48}-\sqrt{75}
Simplify the radicals.16⋅3−25⋅3\sqrt{16}\cdot\sqrt{3}-\sqrt{25}\cdot\sqrt{3}
43−534\sqrt{3}-5\sqrt{3}
Combine the like radicals.−3-\sqrt{3}

Just like we use the Associative Property of Multiplication to simplify 5(3x)5(3x) and get 15x15x, we can simplify 5(3x)5(3\sqrt{x}) and get 15x15\sqrt{x}. We will use the Associative Property to do this in the next example.

13. Example 9.38

Example 9.38

Example 9.38

Simplify: 518−285\sqrt{18}-2\sqrt{8}.

Solution

518−285\sqrt{18}-2\sqrt{8}
Simplify the radicals.5⋅9⋅2−2⋅4⋅25\cdot\sqrt{9}\cdot\sqrt{2}-2\cdot\sqrt{4}\cdot\sqrt{2}
5⋅3⋅2−2⋅2⋅25\cdot{}3\cdot\sqrt{2}-2\cdot{}2\cdot\sqrt{2}
152−4215\sqrt{2}-4\sqrt{2}
Combine the like radicals.11211\sqrt{2}

14. Example 9.39

Example 9.39

Example 9.39

Simplify: 34192−56108\frac{3}{4}\sqrt{192}-\frac{5}{6}\sqrt{108}.

Solution

34192−56108\frac{3}{4}\sqrt{192}-\frac{5}{6}\sqrt{108}
Simplify the radicals.3464⋅3−5636⋅3\frac{3}{4}\sqrt{64}\cdot\sqrt{3}-\frac{5}{6}\sqrt{36}\cdot\sqrt{3}
34⋅8⋅3−56⋅6⋅3\frac{3}{4}\cdot{}8\cdot\sqrt{3}-\frac{5}{6}\cdot{}6\cdot\sqrt{3}
63−536\sqrt{3}-5\sqrt{3}
Combine the like radicals.3\sqrt{3}

15. Example 9.40

Example 9.40

Example 9.40

Simplify: 2348−3412\frac{2}{3}\sqrt{48}-\frac{3}{4}\sqrt{12}.

Solution

2348−3412\frac{2}{3}\sqrt{48}-\frac{3}{4}\sqrt{12}
Simplify the radicals.2316⋅3−344⋅3\frac{2}{3}\sqrt{16}\cdot\sqrt{3}-\frac{3}{4}\sqrt{4}\cdot\sqrt{3}
23⋅4⋅3−34⋅2⋅3\frac{2}{3}\cdot{}4\cdot\sqrt{3}-\frac{3}{4}\cdot{}2\cdot\sqrt{3}
833−323\frac{8}{3}\sqrt{3}-\frac{3}{2}\sqrt{3}
Find a common denominator to subtract thecoefficients of the like radicals.1663−963\frac{16}{6}\sqrt{3}-\frac{9}{6}\sqrt{3}
Simplify.763\frac{7}{6}\sqrt{3}

In the next example, we will remove constant and variable factors from the square roots.

16. Example 9.41

Example 9.41

Example 9.41

Simplify: 18n5−32n5\sqrt{18{n}^{5}}-\sqrt{32{n}^{5}}.

Solution

18n5−32n5\sqrt{18{n}^{5}}-\sqrt{32{n}^{5}}
Simplify the radicals.9n4⋅2n−16n4⋅2n\sqrt{9{n}^{4}}\cdot\sqrt{2n}-\sqrt{16{n}^{4}}\cdot\sqrt{2n}
3n22n−4n22n3{n}^{2}\sqrt{2n}-4{n}^{2}\sqrt{2n}
Combine the like radicals.−n22n-{n}^{2}\sqrt{2n}

17. Example 9.42

Example 9.42

Example 9.42

Simplify: 950m2−648m29\sqrt{50{m}^{2}}-6\sqrt{48{m}^{2}}.

Solution

950m2−648m29\sqrt{50{m}^{2}}-6\sqrt{48{m}^{2}}
Simplify the radicals.925m2⋅2−616m2⋅39\sqrt{25{m}^{2}}\cdot\sqrt{2}-6\sqrt{16{m}^{2}}\cdot\sqrt{3}
9⋅5m⋅2−6⋅4m⋅39\cdot{}5m\cdot\sqrt{2}-6\cdot{}4m\cdot\sqrt{3}
45m2−24m345m\sqrt{2}-24m\sqrt{3}
The radicals are not like and so cannot be combined.

18. Example 9.43

Example 9.43

Example 9.43

Simplify: 28x2−5x32+518x22\sqrt{8{x}^{2}}-5x\sqrt{32}+5\sqrt{18{x}^{2}}.

Solution

28x2−5x32+518x22\sqrt{8{x}^{2}}-5x\sqrt{32}+5\sqrt{18{x}^{2}}
Simplify the radicals.24x2⋅2−5x16⋅2+59x2⋅22\sqrt{4{x}^{2}}\cdot\sqrt{2}-5x\sqrt{16}\cdot\sqrt{2}+5\sqrt{9{x}^{2}}\cdot\sqrt{2}
2⋅2x⋅2−5x⋅4⋅2+5⋅3x⋅22\cdot{}2x\cdot\sqrt{2}-5x\cdot{}4\cdot\sqrt{2}+5\cdot{}3x\cdot\sqrt{2}
4x2−20x2+15x24x\sqrt{2}-20x\sqrt{2}+15x\sqrt{2}
Combine the like radicals.−x2-x\sqrt{2}

19. Section 9.3 Exercises

Section 9.3 Exercises

Section 9.3 Exercises

Add and Subtract Like Square Roots

In the following exercises, simplify.

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

Mixed Practice

Everyday Math

Writing Exercises

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

20. Practice 1

  1. Simplify: 82−928\sqrt{2}-9\sqrt{2}.
  1. Simplify: 53−935\sqrt{3}-9\sqrt{3}.
  1. Simplify: 2x+7x2\sqrt{x}+7\sqrt{x}.
  1. Simplify: 5u+3u5\sqrt{u}+3\sqrt{u}.
  1. Simplify: 7p−6q7\sqrt{p}-6\sqrt{q}.

21. Practice 2

  1. Simplify: 6a−3b6\sqrt{a}-3\sqrt{b}.
  1. Simplify: 411+211+3114\sqrt{11}+2\sqrt{11}+3\sqrt{11}.
  1. Simplify: 610+210+3106\sqrt{10}+2\sqrt{10}+3\sqrt{10}.
  1. Simplify: 55−45+265\sqrt{5}-4\sqrt{5}+2\sqrt{6}.
  1. Simplify: 37−87+253\sqrt{7}-8\sqrt{7}+2\sqrt{5}.

22. Practice 3

  1. Simplify: 7x−77x+47x\sqrt{7x}-7\sqrt{7x}+4\sqrt{7x}.
  1. Simplify: 43y−73y+23y4\sqrt{3y}-7\sqrt{3y}+2\sqrt{3y}.
  1. Simplify: 5xy+45xy−75xy\sqrt{5xy}+4\sqrt{5xy}-7\sqrt{5xy}.
  1. Simplify: 37mn+7mn−47mn3\sqrt{7mn}+\sqrt{7mn}-4\sqrt{7mn}.
  1. Simplify: 18+62\sqrt{18}+6\sqrt{2}.

23. Practice 4

  1. Simplify: 27+43\sqrt{27}+4\sqrt{3}.
  1. Simplify: 32−18\sqrt{32}-\sqrt{18}.
  1. Simplify: 20−45\sqrt{20}-\sqrt{45}.
  1. Simplify: 427−3124\sqrt{27}-3\sqrt{12}.
  1. Simplify: 320−7453\sqrt{20}-7\sqrt{45}.

24. Practice 5

  1. Simplify: 23108−57147\frac{2}{3}\sqrt{108}-\frac{5}{7}\sqrt{147}.
  1. Simplify: 35200−34128\frac{3}{5}\sqrt{200}-\frac{3}{4}\sqrt{128}.
  1. Simplify: 2532−138\frac{2}{5}\sqrt{32}-\frac{1}{3}\sqrt{8}.
  1. Simplify: 1380−14125\frac{1}{3}\sqrt{80}-\frac{1}{4}\sqrt{125}.
  1. Simplify: 32m7−50m7\sqrt{32{m}^{7}}-\sqrt{50{m}^{7}}.

25. Practice 6

  1. Simplify: 27p3−48p3\sqrt{27{p}^{3}}-\sqrt{48{p}^{3}}.
  1. Simplify: 532x2−348x25\sqrt{32{x}^{2}}-3\sqrt{48{x}^{2}}.
  1. Simplify: 748y2−472y27\sqrt{48{y}^{2}}-4\sqrt{72{y}^{2}}.
  1. Simplify: 312x2−2x48+427x23\sqrt{12{x}^{2}}-2x\sqrt{48}+4\sqrt{27{x}^{2}}.
  1. Simplify: 318x2−6x32+250x23\sqrt{18{x}^{2}}-6x\sqrt{32}+2\sqrt{50{x}^{2}}.

26. Practice 7

  1. 82−528\sqrt{2}-5\sqrt{2}
  1. 35+653\sqrt{5}+6\sqrt{5}
  1. 97−1079\sqrt{7}-10\sqrt{7}
  1. 7y+2y7\sqrt{y}+2\sqrt{y}
  1. a−4a\sqrt{a}-4\sqrt{a}

27. Practice 8

  1. 5c+2c5\sqrt{c}+2\sqrt{c}
  1. 8a−2b8\sqrt{a}-2\sqrt{b}
  1. 5m+n5\sqrt{m}+\sqrt{n}
  1. 87+27+378\sqrt{7}+2\sqrt{7}+3\sqrt{7}
  1. 311+211−8113\sqrt{11}+2\sqrt{11}-8\sqrt{11}

28. Practice 9

  1. 33−83+753\sqrt{3}-8\sqrt{3}+7\sqrt{5}
  1. 62+22−356\sqrt{2}+2\sqrt{2}-3\sqrt{5}
  1. 32a−42a+52a3\sqrt{2a}-4\sqrt{2a}+5\sqrt{2a}
  1. 83c+23c−93c8\sqrt{3c}+2\sqrt{3c}-9\sqrt{3c}
  1. 53ab+3ab−23ab5\sqrt{3ab}+\sqrt{3ab}-2\sqrt{3ab}

29. Practice 10

  1. 2pq−5pq+4pq2\sqrt{pq}-5\sqrt{pq}+4\sqrt{pq}
  1. 50+42\sqrt{50}+4\sqrt{2}
  1. 80−35\sqrt{80}-3\sqrt{5}
  1. 27−75\sqrt{27}-\sqrt{75}
  1. 48+27\sqrt{48}+\sqrt{27}

30. Practice 11

  1. 250−3722\sqrt{50}-3\sqrt{72}
  1. 212+3482\sqrt{12}+3\sqrt{48}
  1. 2372+1550\frac{2}{3}\sqrt{72}+\frac{1}{5}\sqrt{50}
  1. 1220−2345\frac{1}{2}\sqrt{20}-\frac{2}{3}\sqrt{45}
  1. 1627−3848\frac{1}{6}\sqrt{27}-\frac{3}{8}\sqrt{48}

31. Practice 12

  1. 1498−13128\frac{1}{4}\sqrt{98}-\frac{1}{3}\sqrt{128}
  1. 72a5−50a5\sqrt{72{a}^{5}}-\sqrt{50{a}^{5}}
  1. 80c7−20c7\sqrt{80{c}^{7}}-\sqrt{20{c}^{7}}
  1. 980p4−698p49\sqrt{80{p}^{4}}-6\sqrt{98{p}^{4}}
  1. 250r8+454r82\sqrt{50{r}^{8}}+4\sqrt{54{r}^{8}}

32. Practice 13

  1. 320x2−445x2+5x803\sqrt{20{x}^{2}}-4\sqrt{45{x}^{2}}+5x\sqrt{80}
  1. 3128y2+4y162−898y23\sqrt{128{y}^{2}}+4y\sqrt{162}-8\sqrt{98{y}^{2}}
  1. 28+68−582\sqrt{8}+6\sqrt{8}-5\sqrt{8}
  1. 175k4−63k4\sqrt{175{k}^{4}}-\sqrt{63{k}^{4}}
  1. 2363−23002\sqrt{363}-2\sqrt{300}

33. Practice 14

  1. 92−829\sqrt{2}-8\sqrt{2}
  1. 813−413−3138\sqrt{13}-4\sqrt{13}-3\sqrt{13}
  1. 80a5−45a5\sqrt{80{a}^{5}}-\sqrt{45{a}^{5}}
  1. 2119−21921\sqrt{19}-2\sqrt{19}
  1. 5627+5848\frac{5}{6}\sqrt{27}+\frac{5}{8}\sqrt{48}

34. Practice 15

  1. 75−108\sqrt{75}-\sqrt{108}
  1. 424x2−54x2+3x64\sqrt{24{x}^{2}}-\sqrt{54{x}^{2}}+3x\sqrt{6}
  1. A decorator decides to use square tiles as an accent strip in the design of a new shower, but she wants to rotate the tiles to look like diamonds. She will use 9 large tiles that measure 8 inches on a side and 8 small tiles that measure 2 inches on a side. Determine the width of the accent strip by simplifying the expression 9(82)+8(22)9(8\sqrt{2})+8(2\sqrt{2}). (Round to the nearest tenth of an inch.)
  1. Explain the difference between like radicals and unlike radicals. Make sure your answer makes sense for radicals containing both numbers and variables.

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