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### Adding & Subtracting Radicals

- Radical expressions can only be added and subtracted if they have the same radicand, and index.
- Some radicals must be re-written before you can add or subtract them. This is done by simplifying the roots first.
- If it is possible to add or subtract to terms with radicals, then we add and subtract their coefficients. This is similar to adding and subtracting like terms.

### Adding & Subtracting Radicals

3√7 + 8√5 − 9√7 − 6√5

- ( 3√7 − 9√7 ) + ( 8√5 − 6√5 )

4√{11} − 20√3 − 16√{11} − 5√3

- ( 4√{11} − 16√{11} ) + ( 20√3 − 5√3 )

64√{17} − 12√2 − 23√2 + 15√{17}

- ( 64√{17} − 15√{17} ) + ( 12√2 − 23√2 )

3√{12x} + 4√{27x} − 5√{3x} + 6√{75x}

- 3√4 ×√{3x} + 4√9 ×√{3x} − 5√1 ×√{3x} + 6√{25} ×√{3x}
- 3 ×2√{3x} + 4 ×3√{3x} − 5 ×1√{3x} + 6 ×5√{3x}
- 6√{3x} + 12√{3x} − 5√{3x} + 30√{3x}

7√{32y} − 8√{72y} + 4√{98y} + 6√{200y}

- 7√{16} ×√{2y} − 8√{36} ×√{2y} + 4√{49} ×√{2y} + 6√{100} ×√{2y}
- 7 ×4√{2y} − 8 ×6√{2y} + 4 ×7√{2y} + 6 ×10√{2y}
- 28√{2y} − 48√{2y} + 28√{2y} + 60√{2y}

5√{196n} − 3√{256n} − 2√{324n} + 8√{36n}

- 5√{49} ×√{4n} − 3√{64} ×√{4n} − 2√{81} ×√{4n} + 8√9 ×√{4n}
- 5 ×7√{4n} − 3 ×8√{4n} − 2 ×9√{4n} + 8 ×3√{4n}

( 4√3 − 6√5 )

^{2}

- ( a − 6 )
^{2}= a^{2}− 2ab + b^{2} - ( 4√3 )
^{2}− 2( 4√3 )( 6√5 ) + ( 6√5 )^{2} - 16 ×3 − ( 2 )( 4 )( 6 )√3 √5 + 36 ×5
- 48 − 48√{3 ×5} + 180

( 2√7 − 3√{11} )

^{2}

- ( 2√7 )
^{2}− 2( 2√7 )( 3√{11} ) + ( 3√{11} )^{2} - 4 ×7 − 2( 2 )( 3 )√7 √{11} + 9 ×11
- 28 − 12√{7 ×11} + 99

( 2√{10} − 4√{12} )( 3√{15} − 5√5 )

- ( 2√{10} )( 3√{15} ) + ( 2√{10} )( − 5√5 ) + ( − 4√{12} )( 3√{15} ) + ( − 4√{12} )( − 5√5 )
- 6√{10} √{15} − 10√{10} √5 − 12√{12} √{15} + 20√{12} √5
- 6√{150} − 10√{50} − 12√{180} + 20√{60}
- 6√{25 ×6} − 10√{25 ×2} − 12√{9 ×4 ×5} + 20√{4 ×15}
- 6 ×5√6 − 10 ×5√2 − 12 ×3 ×2√5 + 20 ×2√{15}

( 6√{40} + 7√{24} )( 10√{18} − 12√{30} )

- ( 6√{40} )( 10√{18} ) + ( 6√{40} )( − 12√{30} ) + ( 7√{24} )( 10√{18} + ) + ( 7√{24} )( − 12√{30} )
- 60√{720} − 72√{1200} + 70√{432} − 84√{720}
- 60√{9 ×16 ×5} − 72√{100 ×4 ×3} + 70√{16 ×9 ×3} − 84√{9 ×16 ×5}
- 60 ×3 ×4√5 − 72 ×10 ×2√3 + 70 ×4 ×3√3 − 84 ×3 ×4√5
- 720√5 − 1440√3 + 840√3 − 1008√5

*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.

Answer

### Adding & Subtracting Radicals

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Objectives 0:07
- Adding and Subtracting Radicals 0:33
- Like Terms
- Bases and Exponents May be Different
- Bases and Powers Must be Same when Adding and Subtracting
- Add Radicals' Coefficients
- Example 1 4:47
- Example 2 6:00
- Adding and Subtracting Radicals Cont. 7:10
- Simplify the Bases to Look the Same
- Example 3 8:23
- Example 4 11:45
- Example 5 15:10

### Algebra 1 Online Course

### Transcription: Adding & Subtracting Radicals

*Welcome back to educator.com.*0000

*In this lesson we are going to take a look at adding and subtracting radicals.*0003

*You may notice in a lot of my other lessons I avoid trying to add or subtract radicals as much as possible.*0009

*It is because there is a few problems that you will run into when you try and these radical expressions.*0015

*Look at some of the problems that we want to avoid.*0020

*Do not get into the actual rule for adding and subtracting that way you will know*0023

*what situations that you can add and subtract and which situations you cannot.*0027

*A lot of the other rules that we have picked up for radicals so far we have been doing lots of multiplication and division.*0035

*You will notice that in those rules they follow pretty much exactly from the rules of exponents.*0042

*There is a nice product rule, quotient rule, and they seem to mimic one another.*0048

*To understand some of the difficult things that we run into with adding, subtracting we have to look at what happens*0053

*when some of our rules for adding, subtracting when you have things with exponents.*0060

*For example, I want to put together x ^{2} + 3x what problems when I run into? can I put those two things together or not?*0064

*You will notice that you will run into quite a bit of a problem.*0075

*These ones do not have the same exponents, I cannot put them together.*0079

*Since the terms are not like terms and they have different exponents maybe we can change it around and try a different situation.*0084

*Let us go ahead and try ab ^{2} + r^{2}.*0100

*Can we put those together? After all the exponents here are exactly the same.*0104

*Can we combine those like terms?*0110

*These one are not like terms.*0114

*We ran into a very similar problem the bases are not the same with these two.*0115

*Those ones are not the same, they do not have the same base then we cannot put them together.*0124

*We usually run into one of those two problems where we are dealing with radicals.*0131

*Either they do not have the same base or they do not have the same exponents.*0135

*Here is a quick example involving radicals so we can see what I’m talking about.*0139

*Here I have 3rd root of x + √3*0142

*If I write those as exponents, then it is like x ^{1}/3 + 3^½.*0147

*These have different bases and they have different exponents, there is no reason why you should be able to put these together.*0153

*After seeing many of these different examples, you might that be under the false assumption that we cannot add any radicals together.*0163

*In some instances you will be able to put these radicals together you have to be very careful on certain conditions.*0173

*One, when putting radicals together you have to make sure that their powers, those would be the indexes of each*0182

*of the radicals are the same and you have to make sure that the radicand or the bases are exactly the same.*0187

*I can put together the 3rd root of 5x with the other 3rd root of x.*0194

*It is completely valid because when written as exponents I have the bases the same and they are both raised to the power of 1/3.*0200

*How would I go about actually putting them together?*0210

*I will treat them just almost like an entire variable.*0213

*If I was adding u + u I will get two u.*0216

*Notice how this common pieces here.*0221

*Another way of saying that is, we simply add together our coefficients out front so one of those should equal two of them.*0224

*Make sure that you keep in mind that if you are going to add and subtract radicals*0237

*you must have the same index on those radicals and you must have the same radicand.*0242

*That is the part underneath radicals.*0246

*Once you get to the addition or subtraction process, look at your coefficients out front, so (5 × √x) + (3 × √x) – (6 × √x) .*0248

*I’m looking at 5 + 3 - 6.*0260

*That will give me a result of 2.*0265

*This is exactly the same process that you would go through if you are just adding like terms.*0270

*That will be u ^{2} + 7u^{2} - u^{2} then you are just looking at these initial coefficients like 2 + 7 - 1, and that would give you the 8.*0275

*Let us see if we can take a look at some examples on when we can add and subtract these radicals.*0288

*In the first one I'm looking at 3 × 4th root of 17 – 4th root of 17*0294

*Let us check, the indexes are exactly the same we are looking at the 4th root and our radicands.*0300

*That is the part underneath, they are both 17.*0308

*We are going to look at the coefficients, 3 - 1 = 2.*0312

*I have 2 4th root of 17 and that one is good.*0318

*Let us look at this other one.*0326

*21√a + 4 3rd root of a*0328

*It is tempting to want to put these ones together but we cannot do it.*0335

*This one is the square root and this one is a cubed root.*0340

*It must have the same index if you have any hope of getting those together.*0353

*Let us look at some others.*0357

*On this one we want to add or subtract if possible.*0361

*I have 3 + √xy + 2 × √xy*0365

*Both of these are dealing with square roots and that is good.*0373

*Both of these are with an xy underneath that root.*0376

*We will simply add together their coefficients.*0381

*This will give us a 5√xy.*0386

*Let us see how that works for the next one.*0392

*7 × 5th root of u ^{3} - 3 × 5th root of y^{3}.*0393

*That is so close.*0400

*Both of them have a 5th root and things are being raised to the third power, all of that is matching up but the variables are completely different.*0402

*One is a u and one is a y.*0412

*Make sure you check your indexes and you check your radicand before you ever put them together using addition or subtraction.*0420

*There are a few instances where the indexes are the same, but it looks like that radicand on the part underneath is completely different.*0431

*It is tempting to write those off and say okay, I probably cannot put those together using addition or subtraction.*0440

*Sometimes if you can do a little bit of simplification and get them the same then you can go ahead and put those together.*0446

*Let me show you an example of numbers.*0453

*Suppose I wanted to put together √2 + √8 and just looking at them I will say that wait 8 and 2 they are not the same, I cannot put them together.*0455

*The √8 over here that is the same as 4 × 2 and I can take √4 and that would leave me with 2 × √2 .*0466

*I can simply rewrite the next one as 2 × √2 .*0480

*In doing so now my radicals are exactly the same and I can simply focus on these coefficients out front.*0485

*I can see that 1 + 2 does equal my 3.*0493

*Do not be afraid to try and simplify these a little bit before you get into the addition or subtraction process.*0497

*Let us try that and keep it in line with these ones.*0504

*We want to rewrite the expressions and then try and add or subtract them if possible.*0507

*The first one I have a -√ 5 + 2 × √125.*0513

*I have -√ 5, 2 and 125 if I want to end up rewriting that, that is a 5 × 25 so that one reduces.*0520

*I have the √5 × 5.*0542

*Let us write that as √ 5 + 10√ 5 and now that I have my radicals the same now just focus on this coefficients -1 + 10 would be 9√ 5.*0548

*The next one I chose a big number but no worries, we can take care of this one.*0568

*We are looking for the 4th root of 3888 + 7 × 4th root of 3*0575

*If I have any hope of putting these together I want to match this 4th root of 3 over here.*0584

*As I go searching for ways to break down that very large number, the very first thing I'm going to try and break it down with is 3.*0591

*Let us see if I can.*0600

*It is the same as 1296 × 3 that is good because if I look at 1296 I can take the 4th root of that and I will get 6.*0608

*I simply have to add together these other radicals here by looking at their coefficients.*0631

*6 + 7 = 13 4th root of 3*0637

*Let us try one more, √72x - √32x*0648

*Let us try and simplify this as much as possible.*0655

*With the first one, looking at 72 is the same as 36 × 2 and with 32 that 16 × 2.*0659

*Notice how I have the square numbers underneath here, but I can go ahead and simplify.*0674

*√36 that will be 6 and I still have that 2 underneath there, √16 will be 4 and there is the 2x for that one.*0679

*I’m looking at 6√2x -4 × √2x or 2 √2x*0691

*These ones are a little bit larger involving some much higher roots, but the same process applies.*0708

*We must get the part underneath the roots the same if we are going to be able to put these two together.*0714

*10 × 4th root of m ^{3} is already broken down as far as it will go.*0724

*The next one I could look at the 6561 and try and take its 4th root and break it apart from its m ^{3}.*0736

*The good news is that one does break down, you will get 9.*0751

*4th root of m ^{3}*0763

*10 4th root of m ^{3} + 9d4th root of m^{3} and we could put those together 10 and 90 is 1004th root of m^{3} .*0767

*Let us try this next one here, this one is the 3rd root of 63xy ^{2} – 3rd root of 125x^{4}y^{5}*0789

*In this one we will not only need to simplify those numbers but also take care of the variables like the x and y.*0799

*First, the numbers the 3rd root of 64 what does that break down into?*0811

*That will go in there 4 times and I have a 3rd root of xy ^{2},*0820

*that one does not break down any further because both of those powers are smaller than the index of 3.*0827

*The 3rd root of 125 would be 5 and let us see what we can do with those variables.*0834

*X ^{3} × x that will be x^{4} and y^{3} × y^{2} that would give me y^{5}.*0843

*This right here I can go ahead and take out of radicals.*0855

*Okay, taking out the x, taking out the y, and I still have xy ^{2}.*0865

*Things are looking good and the part underneath the radical is now exactly the same, we will worry about our coefficients.*0873

*Notice how our coefficients are not like terms, I will be able to write them simply as 4 - 5xy package them together and then write my radical.*0880

*It is like we are just factoring out this common piece and writing it outside here.*0895

*In all cases, make sure you get those radicals exactly the same and combine their coefficients.*0904

*One last example that we can see many of our different rules in action, we will try and combine 3rd root of 2 / x ^{12} - 3 × 3rd root of 3 / x^{15}.*0912

*Starting off, I'm going to use my quotient rule to break that up over the top and over the bottom.*0927

*We will break this one up over the top and over the bottom as well, looks pretty good.*0938

*I will go ahead and simplify the roots on the bottom.*0948

*2 ^{3}rd root of 2 / x^{4} - 3 × 3rd root of 3 / x^{5}.*0953

*Since we are looking to combine things as much as possible,*0967

*I will get a common denominator by putting an x on the bottom and on the top for the left fraction.*0969

*2x 3rd root of 2/x ^{5} - 3 3rd root of 3 /x^{5}.*0979

*As I continue trying to put them together here is one of those situations where we are stuck, we cannot move any farther from there.*0993

*Since this is 3rd root of 2 and this is the 3rd root of 3 and those are different.*1000

*You will know it is tempting but we cannot put them together anymore.*1006

*We will leave this as 2x3rd root of 2 - 33rd root of 3/x ^{5}.*1011

*A lot of different rules to keep track of our radicals but as long as you remember the rules follow directly from the rules for exponents you should be okay.*1021

*Be very careful in adding and subtracting those radicals and make sure everything is satisfied before you even attempt to put them together.*1031

*Thank you for watching educator.com*1040

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Post by Keisha Jordan on October 27, 2015

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