Algebra 2 nth Roots

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In today's lesson, we will be discussing n^th roots.0002

Starting out with the definition of the n^th root: if a and b are real, and n is a positive integer;0007

if aⁿ equals b, then a is an n^th root of b.0014

For example, since 2⁵ = 32, then 2 is the fifth root of 32.0021

And that means that I could rewrite this here: 2 is the fifth root of 32, also rewritten as this, using the radical sign.0037

Or another example--since 3⁶ is 729, 3 is the sixth root of 729.0052

And rewriting this...the sixth root of 729 equals 3.0067

And here, the fifth root of 32 equals 2.0077

So, in earlier lessons in Algebra I, we worked with radicals and worked with square roots.0081

And now, we are going to apply a lot of what we learned to roots other than square roots and go on to learn some new concepts.0087

OK, principal n^th root: this is something we also discussed back in Algebra I, but applying it just to square roots.0099

If there is more than one real n^th root, the non-negative n^th root is called the principal n^th root.0108

OK, so there can be two n^th roots, because thinking about it like this, I just mentioned0116

that the sixth root of 729 is 3; but actually, that is just the principal sixth root of 729,0127

because let's think about the definition: 3⁶ is 729, so according to the definition we just discussed, 3 is a sixth root of 729.0147

Because it is a non-negative number, and there is more than one root, this is the principal sixth root,0163

because -3 to the sixth power is also 729, so this is also a sixth root; -3 is also a sixth root.0170

However, it is not the principal root, because there is another sixth root that is non-negative.0191

The same idea back in Algebra I, when we talked about something like the square root of 4:0201

we would say, "When we use this sign, we mean the principal or non-negative root."0206

So, we would recognize that it is actually 2--this is referring to 2.0210

But in addition, you are aware that 2² is 4, and (-2)² is 4.0215

So, both 2 and -2 were square roots of 4; but when we use this sign, we are talking about the principal, or non-negative, root.0231

if we wanted to refer to -2, then this is the symbolism that we would use: we put a negative sign in front of the square root.0240

And that tells us that we are looking for -2--the same idea here.0251

So, this number right here, n, is called the index; the index right here is 6.0258

Here, the index is just 2, but we don't write it that way; we just leave the 2 off.0268

We know that, when we just have this radical sign with nothing here, then the index is actually 2.0274

Now, a situation can arise that n is odd, and a is less than 0, so a is a negative number.0282

There is no non-negative n^th root: in that case, the principal n^th root is actually negative.0290

So, this is an exception: let's look at the cube root of -8.0296

Well, -2³ is -2 times -2 times -2, which is -8.0301

2³ would be 8; so I only have one cube root of -8, and it is -2.0314

In this case, since that is the only root, this is the principal third, or cube, root of -8.0321

So, if there is more than one root, then you pick the non-negative one; that one is going to be the principal root when you see this symbol, this radical sign.0337

However, if you are in a situation where the only root is a negative number, then that becomes the principal root.0348

Absolute values: again, we talked about this in Algebra I; and let's start out discussing it with square roots,0359

and then realize that it applies to other roots, as well, with an index other than 2.0368

But just starting out talking about square roots: suppose we find the n^th root of an even power.0376

And if the result is an odd power, we must take the absolute value of the result.0382

This guarantees that the answer is non-negative.0388

So, now that we are working with variables, let's look at a situation such as this.0390

Starting out with just the square root of x²: the index here is even--it is 2.0395

And if I take the square root of x², I am going to get x, which is actually x¹.0404

So, my index here is even, and the result is an odd power, since this is really x¹.0410

I will write that there, just for now.0418

In this case, when I take the absolute value of x², I am actually going to take the absolute value of x and put that as my result.0420

And the reason is this: let's say that x equals -3; and I don't know what x is, but let's say that I had a situation where x was -3, and I didn't know it.0428

If I am trying to find x², but x is actually (-3), squared, well, the square root of (-3)²0447

(because really, I have a 2 here, and then these two would just essentially cancel each other out)0462

is -3, but this symbol--this radical sign--tells me I am looking for the principal square root.0466

And this is not the principal square root, because this is also 9, and the principal square root of 9 is 3.0471

If I were to take this and say, "OK, -3² is 9"; the principal square root of that is 3, although another square root of 9 is -3.0481

So, to keep everything consistent--to make sure that I end up with the principal square root,0489

and not another square root--I am going to actually use absolute value bars, so that I will end up with 3.0494

So, this is only for situations where we have an even power, and when we take the root of that power, we end up with an odd power.0503

And the simplest case of that is when you are talking about something like this, taking the square root of x².0515

But the situation would also hold for something like this: ⁶√x¹⁸.0521

Since x¹⁸ equals (x³)⁶ (because remember: when we take a power to a power,0529

then what we do is multiply the exponents: so x³ to the sixth power is really equal to x to the 3 times 6),0541

here I have x³ raised to the sixth power, which is x¹⁸.0552

That means that the sixth root of x¹⁸ is x³.0556

Now, here is the situation where I started out with an even, and it is even; and my result here is an odd power.0564

This is another case where I need to use absolute value bars, because if it turned out,0575

for example, that x was...let's let x equal -2...then x³ = (-2)³ = -8;0580

and I would end up with a negative number here.0596

Now, we don't worry about this when we are talking about even indices that result in an even power.0601

And here is why: let's say that we have something like the sixth root of (-2)¹².0610

Well, this would give me (-2)², because (-2)² to the sixth equals (-2)¹².0625

OK, so the result here is (-2)²; but when I square this, -2 times -2, what I am going to end up with is 4.0646

So, I don't have to worry about the fact that I end up with a negative here, since I have an even power.0666

So, even though this index is even, if my result is also even, I am going to end up with a positive number,0672

because when I take the negative times the negative, it is going to become positive.0680

So, the same holds with variables: even if I have no idea...if I have something like ⁶√y¹²,0686

and I don't know what y is (y could be -2; it could be -4; it could be anything), then I just say,0694

"OK, the sixth root of y¹² is y²," because I know that,0700

no matter what y is, when I multiply it by itself, it is going to become a positive number.0705

So, to sum up: if you find the n^th root of an even power, and the result is an odd power, take the absolute value of the result.0710

So, I had an even index; I ended up with an odd power; I put absolute value bars.0719

I don't worry about that with odd indices; I don't worry about it if I have an even index and I end up with an even power.0726

This is only true for even index resulting in an odd power.0736

In the first example, here I am asked to simplify; and they are asking for the fourth root of y⁸.0744

Well, recall that (y²)⁴ is going to be equal to y⁸, because 2 times 4 is 8.0752

What this tells me is that y² is the fourth root of y⁸.0765

I do not need absolute value bars, because I do have an even index with a result that is an even power.0775

So, no absolute value bars are needed.0789

So, I don't need absolute value bars because, although this is even, this is even as well;0794

so even if it turns out that y is a negative number, like -3, when I take it to the 8^th power, it will become positive.0799

So, I will have my principal root.0807

Example 2: here I have an odd index number: it is the cube root, 3.0810

So again, I don't have to worry about absolute value bars, because I only need to worry about that with even powers that have an odd power result.0816

All right, let's look at what we have here: and think back to earlier lessons with square roots.0825

If I had something like the square root of 18, I could use the product property0834

and say, "OK, this is the same as 9 times 2, which is the same as the square root of 9 times the square root of 2, which equals 3√2."0843

I can do the same thing here: what I am going to do is factor out any numbers that are perfect cubes.0857

So, here I factored out my perfect squares; and I can do the same thing here, but using cubes.0863

81 is actually 3 times 27; and looking at x⁶, this is also a cube; and I am just going to rewrite this,0871

to make it more obvious what is going on here, that x² to the third power is x⁶.0884

I am going to rewrite this like this, to make it more obvious that I have the cube root, which is x².0898

Now, separating out the ones that I can take the cube root of easily: that is 27, and (x²)³.0905

And I am leaving this separate as the cube root of 3.0915

The cube root of 27 is 3; the cube root of (x²)³...well, this and this essentially cancel, and that is going to leave me x².0920

I can't easily find the cube root of 3, so I am just going to leave this as it is.0930

So again, using the same technique that we used with square roots, where we factored out the perfect squares,0934

and then took whatever the square root was and pulled that out--we can do the same thing here.0940

This is a cube; it is actually 3³, so I pulled that out.0945

This is (x²)³, so it is x² cubed, so I can pull out that x²--I can get the cube root.0950

And that leaves me with the cube root of 3 as another factor.0959

Example 3: we are asked to find the square root of 72x⁴y⁶.0968

And I note that I have an even power, so I am going to have to be careful,0975

because I have an even index, and if I end up with an odd power, then I am going to have to use absolute value bars for these variables.0977

So, I have an even index; I have to watch out towards the end here, to make sure I use absolute value bars, if necessary.0985

OK, so let's rewrite this: this is 72x⁴y⁶, and I do have some perfect squares that I can factor out.0993

because 72 is 36 times 2; x⁴ is also a perfect square, because it is (x²)²;1006

and then, y⁶ is actually (y³)².1021

So, x⁴ and y⁶ are also perfect squares.1027

Now, I am going to rewrite this, putting all of the perfect squares together, using the product property.1032

This is 36 times x⁴ times y⁶, times the square root of 2.1036

So, now I am going to find the square root of 36, which I know is 6.1044

The square root of x⁴ is x²; the square root of y⁶, I already said, is y³.1050

And that leaves me with this; however, this is not my final answer, because I had an even index.1057

Here, I have an even power, so that is fine.1063

But y³...that is an odd power, so I am going to put absolute value bars around it,1065

because if it turned out that y is an odd number, like -3, and I took -3 times -3 to get 9, times -3, and that would be -27.1069

And that isn't what I want, because, when I see this radical symbol, they are looking for the principal root, and that would be a positive number.1079

So, just to ensure that I am working with a positive number here, I need to use absolute value bars, only around y³.1088

In this last example, we are looking for the cube root of -z¹²w⁶.1099

So, this is an odd index, so right away I know I don't have to worry about absolute value bars.1106

So, let's look first at this w⁶, because I don't have to worry about a negative sign with that, so that is simpler.1111

Well, if I am looking for the cube root, I need w to the something to the third power, that equals w⁶.1122

And since 2 times 3 is 6, I know that the cube root of w⁶ is w².1130

So, that handles that; now, looking at z¹², I need z to the something that, when raised to the third power, is going to give me z¹².1141

Well, 4 times 3 is 12; this is 4 times 3; z to the 4 times 3 equals z to the 12.1155

Now, there is a negative in front of that, but I can just think of that as -1.1162

And I know that -1 cubed is -1, times -1 is 1, times -1 is -1.1167

So, the cube root of -1 is -1; so I can look at this just separately--this is the cube root of -1, times z¹², times w⁶.1178

And I found each cube root, right in here; I could even rewrite it that way to make it clear what is happening.1191

This would be -1 cubed, times z⁴, cubed, times y², cubed.1200

So, when I take the cube root of each of these, I just essentially cancel these; and I am going1211

to end up with -1 times z⁴ times w², or just -z⁴w².1216

Again, with an odd index, I don't need to worry about absolute value bars.1228

So, the easiest way is just to approach each section individually, look for the cube root, and then combine that all in your final answer.1231

That concludes this lesson on n^th roots.1242

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