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You add and subtract radicals

In algebra, we can combine terms that are similar eg. 2a + 3a = 5a 8x2 + 2x - 3x2 = 5x2 + 2x Similarly for surds, we can combine those that are similar. They must have the same radicand (number under the radical) and the same index (the root that we are taking). (a) 27 - 57 + 7 Answer In this question, the radicand (the number under the square root) is 7 in each item, and the index is 2 (that is, we are taking square root) in each item, so we can add and subtract the like terms as follows: 27 - 57 + 7 = -27 What I did (in my head) was to factor out 7 as follows: 27 - 57 + 7 = (2 - 5 + 1)7 = -27 (b) `root(5)6+4root(5)6-2root(5)6` Answer Once again, each item has the same radicand (`6`) and the same index (`5`), so we can collect like terms as follows: `root(5)6+4root(5)6-2root(5)6=3root(5)6` (c) `sqrt5+2sqrt3-5sqrt5` Answer In this example, the like terms are the 5 and -5 (same radicand, same index), so we can add them, but the 3 term has a different radicand and so we cannot do anything with it. `sqrt5+2sqrt3-5sqrt5=2sqrt3-4sqrt5` Example 2 (a) `6sqrt7-sqrt28+3sqrt63` Answer In each part, we are taking square root, but the number under the square root is different. We need to simplify the radicals first and see if we can combine them. We recognize that there is a 7 term involved in each item. `6sqrt(7) sqrt(28) + 3sqrt(63) ` `= 6sqrt(7) - sqrt(4 times 7) + 3sqrt(9 times 7)` `= 6sqrt(7) - 2sqrt(7) + 3(3 sqrt(7))` `= 6sqrt(7) - 2sqrt(7) + 9sqrt(7)` `=13sqrt(7)` (b) `3sqrt125-sqrt20+sqrt27` Answer We simplify the radicals first and then collect together like terms. `3sqrt(125) - sqrt(20) + sqrt(27)` `= 3sqrt(25 times 5) - sqrt(4 times 5) + sqrt(9 times 3)` `=3(5sqrt(5)) - 2 sqrt(5) + 3sqrt(3)` `=13sqrt(5) + 3sqrt(3)` Example 3 Simplify: `sqrt(2/(3a))-2sqrt(3/(2a)` Answer Our aim here is to remove the radicals from the denominator of each fraction and then to combine the terms into one expression. First, we multiply top and bottom of each fraction with their respective denominators. This gives us a perfect square in the denominator in each case, and we can remove the radical. `sqrt(frac{2}{3a}) - 2 sqrt(frac{3}{2a})` ` = sqrt(frac{2(3a)}{3a(3a)}) - 2 sqrt(frac{3(2a)}{2a(2a)})` ` = sqrt(frac{6a}{9a^2}) - 2sqrt(frac{6a}{4a^2})` We then simplify and see that we have like terms (`sqrt(6a)`). `= frac{1}{3a}sqrt(6a) - frac{2}{2a}sqrt(6a)` `=frac{1}{3a} sqrt(6a) - frac{1}{a} sqrt(6a)` We then proceed to subtract the fractions by finding a common denominator (`3a`). `= frac{sqrt(6a) - 3sqrt(6a)}{3a}` `=frac{-2sqrt(6a)}{3a}` `=-frac{2}{3a} sqrt(6a)` Exercises Q1 `sqrt7+sqrt63` Answer `sqrt(7) + sqrt(63)` ` = sqrt(7) + sqrt(9 times 7)` `=sqrt(7) + 3sqrt(7)` `=4sqrt(7)` Q2 `2sqrt44sqrt99+sqrt2sqrt88` Answer `2sqrt(44) - sqrt(99) + sqrt(2) sqrt(88)` `=2sqrt(4 times 11) - sqrt(9 times 11) +` ` sqrt(2) sqrt(4 times 2 times 11)` `=2(2)sqrt(11) - (3)sqrt(11) +` ` sqrt(2)(2)sqrt(2)sqrt(11)` `=4sqrt(11) - 3sqrt(11) + 4sqrt(11)` `=5sqrt(11)` Q3 `root(6)sqrt2-root(12)(2^13)` Answer This looks ugly, but don't panic. For the first item, finding the 6th root of a square root is the same as finding the 12th root. We need to use this rule from before: `rootmrootn(a) = root(mn)(a)` So the first term is: `root(6)(sqrt(2)) = root(6 times 2)(2) = root(12)(2)` For the second term, we need to split up the `2^13` as follows: `2^13= 2^12? 2` We do this so that we end up with a "12th root of 2" term so that we can simplify the final answer. We also need the following identity for this part: `root(n)(a^n) = ( rootn(a))^n = rootn((a^n)) = a` We'll use this in the second line, to simplify the 12th root of `2^12`: `root(12)(2^13)` ` = root(12)(2^12 times 2)` `=root(12)(2^12) times root(12)(2)` `=2 root(12)(2)` Now let's put this all together and get the final answer: `root(6)sqrt(2) - root(12)(2^13)` ` = root(12)(2) - 2 root(12)(2)` `= - root(12)(2)` Radicals can be simplified through adding and subtracting, but you should keep in mind that you sometimes can't "cleanly" simplify square roots down into a number. The first thing to note is that radicals can only be added and subtracted if they have the same root number. If the radicals in a question are unlike, you won't be able to combine them together. That's one of the main rules for radicals that you should remember. How to subtract radicals Our example will help you grasp the basics of what we mean when we say you have to find like radicals before combining them. The simplifying radicals problem below deals with how to subtract radicals. Question 1: 90-10\sqrt{90}-\sqrt{10}90-10 Solution: We are going to first simplify this expression. We will perform a division analysis for the number 90 in order to identify its roots: the result of division analysis for 90 We learn that 90\sqrt{90}90 can be re-written as: 5233\sqrt{5 \cdot 2 \cdot 3 \cdot 3}5233 Since we are dealing with square roots (which is what a radical sign stands for. There should be a tiny 2 next to the radical sign to indicate that we're working with square roots, but it is generally accepted to simply write the sign without it), we are looking for numbers in pair. In this case, we can see that we have a pair of 3s Now, we can take the pair outside of the radical (square root), leaving us with: 352=3103 \sqrt{5 \cdot 2} = 3 \sqrt{10}352=310 This means our question can be transformed into the following expression: 310-103 \sqrt{10} - \sqrt{10}310-10 You can see that we've now identified the "like" radicals in the question: . We can finally do the subtraction, and the result gives us: 2102 \sqrt{10}210 steps of subtracting radicals As you can see, the number that we ultimately get isn't a clean, whole number, but rather a number followed by a radical. It's best to leave the number like this because if we take the square root of the radical, we'll get a messy decimal number. Unless if the question tells you to, there's no need to simplify it further. You've already got the simplest radical form via this answer. How to add radicals In this example, let's take a look to see how to add radicals, as well as deal with a cubic root rather than a square root. Question 2: Combine the expression into a single radical. 340+3135{^3}\sqrt{40}+{^3}\sqrt{135}340+3135 Solution: Again, let us first perform a division analysis for 340{^3}\sqrt{40}340 and 3135{^3}\sqrt{135}3135 340{^3}\sqrt{40}340: the result of division analysis for 40 3135{^3}\sqrt{135}3135: the result of division analysis for 135 This allows us to rewrite the original question into this expression: 35222+35333{^3}\sqrt{5 \cdot 2 \cdot 2 \cdot 2} + {^3}\sqrt{5 \cdot 3 \cdot 3 \cdot 3}35222+35333 Since we are dealing with cube root, we are looking for numbers in triplets. In this case, we have a triplet of 2s in and a triplet of 3s in Now, we can take the triplets outside of the radical (cubic root), and get this expression: taking out the triplets of 2 and 3 out from the cubic root Now that the two numbers have the same radical: We can do the addition, and the result is: 5355{^3}\sqrt{5}535 steps of adding radicals Dealing with other roots of varying powers can be tackled in the same way. Just make sure you're combing the same roots, and not, say, a cubic root with a square root when you want to simplify radicals. There are some examples here that illustrate what to do when you have different roots. By the end of this section, you will be able to: Add and subtract like square roots Add and subtract square roots that need simplification Before you get started, take this readiness quiz. Add: 3x+9x3x+9x 5m+5n5m+5n. If you missed this problem, review Example 1.24. Simplify: 50x350x3. If you missed this problem, review Example 9.16. 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+72+7 in this way: 2+7Add inside the radical.9Simplify.32+7Add inside the radical.9Simplify.3 So if we have to add 2+72+7, we must not combine them into one radical. 2+72+72+72+7 Trying to add square roots with different radicands is like trying to add unlike terms. But, just like we can addx+x,we can add3+3.x+x=2x3+3=23But, just like we can addx+x,we can add3+3.x+x=2x3+3=23 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+8x3x+8x and the result is 11x.11x. 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. 22-7222-72 Since the radicals are like, we subtract the coefficients. -52-52 3y+4y3y+4y Since the radicals are like, we add the coefficients. 7y7y 4x-2y4x-2y Since the radicals are not like, we cannotsubtract them. We leave the expression as is. 4x-2y4x-2y Simplify: 513+413+213513+413+213. 513+413+213513+413+213 Since the radicals are like, we add the coefficients. 11131113 Simplify: 411+211+311411+211+311. Simplify: 610+210+310610+210+310. Simplify: 26-66+3326-66+33. 26-66+3326-66+33 Since the first two radicals are like, wesubtract their coefficients. -46+33-46+33 Simplify: 55-45+2655-45+26. Simplify: 37-87+2537-87+25. Simplify: 25n-65n+45n25n-65n+45n. 25n-65n+45n25n-65n+45n Since the radicals are like, we combine them. 05n05n Simplify. 0 Simplify: 7x-77x+47x7x-77x+47x. Simplify: 43y-73y+23y43y-73y+23y. When radicals contain more than one variable, as long as all the variables and their exponents are identical, the radicals are like. Simplify: 3xy+53xy-43xy3xy+53xy-43xy. 3xy+53xy-43xy3xy+53xy-43xy Since the radicals are like, we combine them. 23xy23xy Simplify: 5xy+45xy-75xy5xy+45xy-75xy. Simplify: 37mn+7mn-47mn37mn+7mn-47mn. 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. 20+3520+35 Simplify the radicals, when possible. 4?5+354?5+35 25+3525+35 Combine the like radicals. 5555 48-7548-75 Simplify the radicals. 16?3-25?316?3-25?3 43-5343-53 Combine the like radicals. -3-3 Just like we use the Associative Property of Multiplication to simplify 5(3x)5(3x) and get 15x15x, we can simplify 5(3x)5(3x) and get 15x15x. We will use the Associative Property to do this in the next example. 518-28518-28 Simplify the radicals. 5?9?2-2?4?25?9?2-2?4?2 5?3?2-2?2?25?3?2-2?2?2 152-42152-42 Combine the like radicals. 112112 Simplify: 427-312427-312. Simplify: 320-745320-745. Simplify: 34192-5610834192-56108. 34192-5610834192-56108 Simplify the radicals. 3464?3-5636?33464?3-5636?3 34?8?3-56?6?334?8?3-56?6?3 63-5363-53 Combine the like radicals. 33 Simplify: 23108-5714723108-57147. Simplify: 35200-3412835200-34128. Simplify: 2348-34122348-3412. 2348-34122348-3412 Simplify the radicals. 2316?3-344?32316?3-344?3 23?4?3-34?2?323?4?3-34?2?3 833-323833-323 Find a common denominator to subtract thecoefficients of the like radicals. 1663-9631663-963 Simplify. 763763 Simplify: 2532-1382532-138. Simplify: 1380-141251380-14125. In the next example, we will remove constant and variable factors from the square roots. Simplify: 18n5-32n518n5-32n5. 18n5-32n518n5-32n5 Simplify the radicals. 9n4?2n-16n4?2n9n4?2n-16n4?2n 3n22n-4n22n3n22n-4n22n Combine the like radicals. -n22n-n22n Simplify: 32m7-50m732m7-50m7. Simplify: 27p3-48p327p3-48p3. Simplify: 950m2-648m2950m2-648m2. 950m2-648m2950m2-648m2 Simplify the radicals. 925m2?2-616m2?3925m2?2-616m2?3 9?5m?2-6?4m?39?5m?2-6?4m?3 45m2-24m345m2-24m3 The radicals are not like and so cannot be combined. Simplify: 532x2-348x2532x2-348x2. Simplify: 748y2-472y2748y2-472y2. Simplify: 28x2-5x32+518x228x2-5x32+518x2. 28x2-5x32+518x228x2-5x32+518x2 Simplify the radicals. 24x2?2-5x16?2+59x2?224x2?2-5x16?2+59x2?2 2?2x?2-5x?4?2+5?3x?22?2x?2-5x?4?2+5?3x?2 4x2-20x2+15x24x2-20x2+15x2 Combine the like radicals. -x2-x2 Simplify: 312x2-2x48+427x2312x2-2x48+427x2. Simplify: 318x2-6x32+250x2318x2-6x32+250x2. Access this online resource for additional instruction and practice with the adding and subtracting square roots. Adding/Subtracting Square Roots Section 9.3 Exercises Add and Subtract Like Square Roots In the following exercises, simplify. 163. 164. 169. 170. 11b-511b+311b11b-511b+311b 171. 172. 173. 53ab+3ab-23ab53ab+3ab-23ab 174. 811cd+511cd-911cd811cd+511cd-911cd 175. 176. 112rs-92rs+32rs112rs-92rs+32rs Add and Subtract Square Roots that Need Simplification In the following exercises, simplify. 194. 195. 201. 202. 203. 204. 205. 320x2-445x2+5x80320x2-445x2+5x80 206. 228x2-63x2+6x7228x2-63x2+6x7 207. 3128y2+4y162-898y23128y2+4y162-898y2 208. 375y2+8y48-300y2375y2+8y48-300y2 Mixed Practice 211. 212. 217. 218. 227. 424x2-54x2+3x6424x2-54x2+3x6 228. 229. 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(82)+8(22). (Round to the nearest tenth of an inch.) 230. Suzy wants to use square tiles on the border of a spa she is installing in her backyard. She will use large tiles that have area of 12 square inches, medium tiles that have area of 8 square inches, and small tiles that have area of 4 square inches. Once section of the border will require 4 large tiles, 8 medium tiles, and 10 small tiles to cover the width of the wall. Simplify the expression 412+88+104412+88+104 to determine the width of the wall. (Round to the nearest tenth of an inch.) 231. Explain the difference between like radicals and unlike radicals. Make sure your answer makes sense for radicals containing both numbers and variables. 232. Explain the process for determining whether two radicals are like or unlike. Make sure your answer makes sense for radicals containing both numbers and variables. 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?

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