nth Roots
- The principal nth root is the nonnegative root. Exception: If n is odd and the radicand is negative, the principal nth root is negative.
- If the nth root of an even power is an odd power, take the absolute value of the result to guarantee that the result is nonnegative.
nth Roots
- Rewrite the power of 16 as products of power of 8 as much as possible.
- Rewrite the power of 20 as products of power 8 as much as possible.
- Rewrite the power of 12 as products of powers 3 as much as possible and also
- find the a number cubed that equals 125.
- ^{3}√{125x^{12}} = ^{3}√{5*5*5*x^{3}*x^{3}*x^{3}*x^{3}} = ^{3}√{5^{3}*x^{3}*x^{3}*x^{3}*x^{3}} = ^{3}√{5^{3}}*^{3}√{x^{3}}*^{3}√{x^{3}}*^{3}√{x^{3}}*^{3}√{x^{3}}
- Simplify
- Rewrite the power of 8 as products of powers 4 as much as possible and also
- find the a number that when raised to the 4th power equals 81.
- ^{4}√{81x^{10}} = ^{4}√{9*9*x^{4}*x^{4}*x^{2}} = ^{4}√{3*3*3*3*x^{4}*x^{4}*x^{2}} = ^{4}√{3*3*3*3}*^{4}√{x^{4}}*^{4}√{x^{4}}*^{4}√{x^{2}}
- Simplify
- Rewrite the power of 8 as products of powers 4 as much as possible and also
- find the a number that when raised to the 4th power equals 81.
- ^{4}√{81x^{10}} = ^{4}√{9*9*x^{4}*x^{4}*x^{2}} = ^{4}√{3*3*3*3*x^{4}*x^{4}*x^{2}} = ^{4}√{3*3*3*3}*^{4}√{x^{4}}*^{4}√{x^{4}}*^{4}√{x^{2}}
- Simplify
- Rewrite the powers of 8 and 12 as products of powers 2 as much as possible and also
- Find the prime factorization of 96 to look for numbers to get out of the radical sign.
- 96 =
- 96 = 2*2*2*2*2*3
- √{96x^{8}y^{12}} = √{2*2*2*2*2*3*x^{2}*x^{2}*x^{2}*x^{2}*y^{2}*y^{2}*y^{2}*y^{2}*y^{2}*y^{2}}
- √{2*2} *√{2*2} *√{2*3} *√{x^{2}} *√{x^{2}} *√{x^{2}} *√{x^{2}} *√{y^{2}} *√{y^{2}} *√{y^{2}} *√{y^{2}} *√{y^{2}} *√{y^{2}}
- Simplify
- 2*2*√{2*3} *x*x*x*x*y*y*y*y*y*y
- Rewrite the powers of 8 and 12 as products of powers 2 as much as possible and also
- Find the prime factorization of 96 to look for numbers to get out of the radical sign.
- 40 =
- 40 = 2*2*2*5
- √{40x^{4}y^{6}} = √{2*2*2*5*x^{2}*x^{2}*y^{2}*y^{2}*y^{2}}
- Simplify
- = √{2*2} *√{2*5} *√{x^{2}} *√{x^{2}} *√{y^{2}} *√{y^{2}} *√{y^{2}}
- Rewrite the powers of 8 and 12 as products of powers 2 as much as possible and also
- Find the prime factorization of 32 to look for numbers to get out of the radical sign.
- 32 =
- 32 = 2*2*2*2*2
- √{32x^{5}y^{7}} = √{2*2*2*2*2*x^{2}*x^{2}*x*y^{2}*y^{2}*y^{2}*y}
- Simplify
- = √{2*2} *√{2*2} *√2 *√{x^{2}} *√{x^{2}} *√x *√{y^{2}} *√{y^{2}} *√{y^{2}} *√y
- Rewrite the powers of 3 and 4 and 5 as products of powers 2 as much as possible and also
- Find the prime factorization of 56 to look for numbers to get out of the radical sign.
- 56 =
- 56 = 2*2*2*7
- √{56x^{3}y^{4}z^{5}} = √{2*2*2*7*x^{2}*x*y^{2}*y^{2}*z^{2}*z^{2}*z}
- √{2*2} *√{2*7} *√{x^{2}} *√x *√{y^{2}} *√{y^{2}} *√{z^{2}} *√{z^{2}} *√z
- Simplify
- 2*√{2*7} *x*√x *y*y*√y *z*z*√z
- 2xy^{2}√{14xyz}
- Rewrite the powers of 8 and 12 as products of powers 2 as much as possible and also
- Find the prime factorization of 96 to look for numbers to get out of the radical sign.
- 40 =
- 40 = 2*2*2*5
- √{40x^{4}y^{6}} = √{2*2*2*5*x^{2}*x^{2}*y^{2}*y^{2}*y^{2}}
- Simplify
- = √{2*2} *√{2*5} *√{x^{2}} *√{x^{2}} *√{y^{2}} *√{y^{2}} *√{y^{2}}
- Rewrite the powers of 8 and 12 as products of powers 2 as much as possible and also
- Find the prime factorization of 32 to look for numbers to get out of the radical sign.
- 32 =
- 32 = 2*2*2*2*2
- √{32x^{5}y^{7}} = √{(2^{2})^{2}*2*( x^{2} )^{2}*x*( y^{3} )^{2}y*( z^{4} )^{2}}
- Simplify
- √{(2^{2})^{2}*2*( x^{2} )^{2}*x*( y^{3} )^{2}y*( z^{4} )^{2}} = √{(2^{2})^{2}*2} *√{( x^{2} )^{2}*x} *√{( y^{3} )^{2}y} *√{( z^{4} )^{2}}
- 4√2 *x^{2}√x *y^{3}√y *z^{4}
*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
nth Roots
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
- Definition of the nth Root 0:07
- Example: 5th Root
- Example: 6th Root
- Principal nth Root 1:39
- Example: Principal Roots
- Using Absolute Values 5:58
- Example: Square Root
- Example: 6th Root
- Example: Negative
- Example 1: Simplify Radicals 12:23
- Example 2: Simplify Radicals 13:29
- Example 3: Simplify Radicals 16:07
- Example 4: Simplify Radicals 18:18
Algebra 2
Transcription: nth Roots
Welcome to Educator.com.0000
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 a^{n} equals b, then a is an n^{th} root of b.0014
For example, since 2^{5} = 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 3^{6} 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: 3^{6} 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 2^{2} 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, -2^{3} is -2 times -2 times -2, which is -8.0301
2^{3} 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 x^{2}: the index here is even--it is 2.0395
And if I take the square root of x^{2}, I am going to get x, which is actually x^{1}.0404
So, my index here is even, and the result is an odd power, since this is really x^{1}.0410
I will write that there, just for now.0418
In this case, when I take the absolute value of x^{2}, 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 x^{2}, but x is actually (-3), squared, well, the square root of (-3)^{2}0447
(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, -3^{2} 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 x^{2}.0515
But the situation would also hold for something like this: ^{6}√x^{18}.0521
Since x^{18} equals (x^{3})^{6} (because remember: when we take a power to a power,0529
then what we do is multiply the exponents: so x^{3} to the sixth power is really equal to x to the 3 times 6),0541
here I have x^{3} raised to the sixth power, which is x^{18}.0552
That means that the sixth root of x^{18} is x^{3}.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 x^{3} = (-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}√y^{12},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 y^{12} is y^{2}," 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 y^{8}.0744
Well, recall that (y^{2})^{4} is going to be equal to y^{8}, because 2 times 4 is 8.0752
What this tells me is that y^{2} is the fourth root of y^{8}.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 x^{6}, 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 x^{2} to the third power is x^{6}.0884
I am going to rewrite this like this, to make it more obvious that I have the cube root, which is x^{2}.0898
Now, separating out the ones that I can take the cube root of easily: that is 27, and (x^{2})^{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 (x^{2})^{3}...well, this and this essentially cancel, and that is going to leave me x^{2}.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 3^{3}, so I pulled that out.0945
This is (x^{2})^{3}, so it is x^{2} cubed, so I can pull out that x^{2}--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 72x^{4}y^{6}.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 72x^{4}y^{6}, and I do have some perfect squares that I can factor out.0993
because 72 is 36 times 2; x^{4} is also a perfect square, because it is (x^{2})^{2};1006
and then, y^{6} is actually (y^{3})^{2}.1021
So, x^{4} and y^{6} 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 x^{4} times y^{6}, 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 x^{4} is x^{2}; the square root of y^{6}, I already said, is y^{3}.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 y^{3}...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 y^{3}.1088
In this last example, we are looking for the cube root of -z^{12}w^{6}.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 w^{6}, 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 w^{6}.1122
And since 2 times 3 is 6, I know that the cube root of w^{6} is w^{2}.1130
So, that handles that; now, looking at z^{12}, I need z to the something that, when raised to the third power, is going to give me z^{12}.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 z^{12}, times w^{6}.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 z^{4}, cubed, times y^{2}, 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 z^{4} times w^{2}, or just -z^{4}w^{2}.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
Thanks for visiting Educator.com.1245
0 answers
Post by julius mogyorossy on October 22, 2014
Dr. Carleen, you say that when deciding whether to use absolute value bars to take in to account the index, this seems to contradict Educator, it seems unlogical, what's up? The un is intentional, long story.
1 answer
Last reply by: Kavita Agrawal
Mon Jun 17, 2013 10:41 PM
Post by Jose Gonzalez-Gigato on January 5, 2012
In Example I, the result should be y^2, not y^8.
1 answer
Last reply by: Hoochie Mamma
Tue Aug 23, 2011 10:16 PM
Post by Hoochie Mamma on August 21, 2011
I hate that you can't point and click where you want the volume to be set. some clips are loud and others are softer and I'm constantly trying to adjust the volume.
Also, the same with the rewind feature. You can't point and click wheere you want the video to play. If you missed one word and try to rewind a little, you can't. You're forced to rewind back 30 seconds and longer. When watching an hour long video and rewinding often, it ends up taking forever to have to rewind so far back.