Algebra 1 Adding & Subtracting Radicals

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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 know0023

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 happens0053

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

For example, I want to put together x² + 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 ab² + r².0100

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 + √30142

If I write those as exponents, then it is like x¹/3 + 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 each0182

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 radicals0237

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 u² + 7u² - u² 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 170294

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 a0328

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 × √xy0365

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 u³ - 3 × 5th root of y³.0393

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 30575

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 30637

Let us try one more, √72x - √32x0648

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 √2x0691

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 m³ 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 m³.0736

The good news is that one does break down, you will get 9.0751

4th root of m³0763

10 4th root of m³ + 9d4th root of m³ and we could put those together 10 and 90 is 1004th root of m³ .0767

Let us try this next one here, this one is the 3rd root of 63xy² – 3rd root of 125x⁴y⁵0789

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 xy²,0820

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

X³ × x that will be x⁴ and y³ × y² that would give me y⁵.0843

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 xy².0865

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 / x¹² - 3 × 3rd root of 3 / x¹⁵.0912

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

2³rd root of 2 / x⁴ - 3 × 3rd root of 3 / x⁵.0953

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/x⁵ - 3 3rd root of 3 /x⁵.0979

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/x⁵.1011

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

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