**Урок №4. Корінь n-го степеня. Арифметичний корінь n-го степеня, його властивості (10 клас. Алгебра) — Transcript & Summary | SozAI**
Source: https://sozai.app/transcript/root-nth-degree-arithmetic-properties-algebra/

This 10th grade algebra lesson explains the n-th root, arithmetic n-th root, and their properties with examples and practice problems.

## Key Takeaways

- The n-th root is the inverse operation of exponentiation for natural number indices.
- Even-degree roots of positive numbers have two values; odd-degree roots have one value for all real numbers.
- Roots of products and quotients can be separated into roots of individual factors.
- Arithmetic n-th roots are defined only for non-negative radicands and yield non-negative results.
- Simplifying nested roots involves multiplying the root indices and adjusting powers accordingly.

## What the video covers

- Introduction to the concept of the n-th root where n is a natural number greater than one.
- Definition of the n-th root as a number whose n-th power equals the radicand.
- Explanation of even-degree roots having two values (positive and negative) for positive numbers and no roots for negative numbers.
- Explanation of odd-degree roots having exactly one value for any real number, positive or negative.
- Properties of roots including product and quotient rules for nth roots.
- Simplification of nested roots and changing root indices by multiplying powers.
- Definition and properties of the arithmetic n-th root, which is always non-negative.
- Examples demonstrating calculation of roots and simplification of expressions.
- Encouragement for viewer interaction through likes, comments, and participation in tests.
- Acknowledgment of channel supporters and sponsors.

## Chapters

1. 00:00 Introduction and Lesson Overview
2. 01:22 Definition of the n-th Root and Natural Numbers
3. 02:38 Even-Degree Roots and Their Properties
4. 04:05 Odd-Degree Roots and Their Properties
5. 05:14 Properties of Roots: Product and Quotient Rules
6. 06:36 Simplifying Nested Roots and Changing Indices
7. 07:37 Arithmetic n-th Root Definition and Properties
8. 09:17 Examples and Practice Problems
9. 19:11 Summary and Channel Supporters

Answers

## Questions about this video

What is the n-th root of a number?

The n-th root of a number 'a' is a number which, when raised to the power n, equals 'a'. Here, n is a natural number greater than one.

Can you find the even-degree root of a negative number?

No, the even-degree root of a negative real number does not exist because no real number raised to an even power results in a negative number.

How do odd-degree roots differ from even-degree roots?

Odd-degree roots exist for all real numbers and yield one value, which is positive if the number is positive and negative if the number is negative, unlike even-degree roots.

## Full Transcript — Download SRT & Markdown

00:00

Speaker A

Video lesson created with the support of the website testic.com.ua, helping psychology students process psychological research and testing for large groups of people.

00:26

Speaker A

Welcome to the Top School channel. As always, I am your host, Oleksandr Borysovych. And today we have another video lesson for the 10th grade in algebra, and the topic is: the n-th root, the arithmetic n-th root, and its

00:40

Speaker A

properties. Before we start the lesson, I have a small request for you. Please hit the like button for this video and subscribe to the Top School channel if you haven't already. Subscribed and liked. So, let's begin. A little bit of

00:58

Speaker A

theory, after which we will look at examples, and for dessert at the end, tests, for which we will receive a grade. So, what is a root? The n-th root, where n is a natural number, and n is greater than one, of a number 'a'

01:13

Speaker A

is a number whose n-th power equals 'a'. I draw your attention to the fact that we will be talking about roots.

01:22

Speaker A

Where n is a natural number, that is, one formed by counting: 1, 2, 3, 4. We will not be talking about cases where the root index is a fractional number, a decimal, or a fraction. We are currently talking about an integer.

01:37

Speaker A

Perhaps you already know from previous grades what a square root or a cube root is. So today, we will expand those skills. We will generally talk about the n-th root. The expression "n-th root of a" is called the n-th root of

01:53

Speaker A

number 'a', where 'a' is the radicand, and the root sign—sometimes also called a radical, a word you are already familiar with. And 'n' is the root index. What are its properties?

02:09

Speaker A

Let's examine them in detail. The n-th root of the number 0 equals zero. Well, this is logical, since zero to any power will always be zero. So the record, the n-th root of zero equals zero. Yes. Yes. There is also the

02:25

Speaker A

property of the even-degree root. Let's recall what even numbers are: 2, 4, 6, 8. Of a positive real number, it has two values that are the same in absolute value but opposite in sign.

02:38

Speaker A

Let me explain with an example. The 4th root of 81 = 3, and the 4th root of 81 = -3. Two values. We are finding an even-degree root. Four is an even number. Of a positive real number, in

02:55

Speaker A

this case, 81. And we get two values, 3 and -3. The same in absolute value, opposite in sign. Why is that? Well, because 3 to the 4th power is 81, and -3 to the 4th power is also 81. You

03:12

Speaker A

understand what we are talking about, right? We had the operation of raising to a power, and the root is the inverse operation to exponentiation. The even-degree root of a negative real number does not exist. We just looked at positive numbers, now let's talk

03:31

Speaker A

about negative ones. I will explain. No. Let's take the number -81 and try to find its fourth root. It doesn't exist. Why? Because there is no number that, when raised to the fourth power, would equal -81. We looked at examples

03:50

Speaker A

above. 3 and -3 raised to the fourth power both give 81, and it will never be -81. That is why an even root of a negative number also does not exist.

04:05

Speaker A

Odd degree root. I draw your attention to the fact that those were even degrees, this is an odd one. For any real number, it has one value. That means you can find an odd root for both a positive and a negative number.

04:19

Speaker A

Moreover, it will be a positive number if the number is positive, and negative if it is negative. The cube root of 8 is 2, since 2 to the power of 3 equals 8. See, you can find the root of eight

04:34

Speaker A

without any problems. And you can also find the cube root of -8. It equals -2, because -2 to the power of 3 equals -8.

04:42

Speaker A

So, everything is much simpler with odd roots. A root can always be extracted there. But with even roots, there are limitations. It doesn't exist for a negative number, but it exists for a positive one. Did you remember that?

04:55

Speaker A

Great. Let's move on. What are the properties of roots? If I have a product of numbers a and b, and I need to find their nth root. Sometimes it happens that the root of a product is impossible or difficult to find, so I

05:14

Speaker A

can find the roots of each number separately. Well, and then multiply them. The nth root of a times the nth root of b. And this formula also works in the other direction. If you have two numbers from which roots cannot be

05:29

Speaker A

extracted, you can put them under a common radical sign. Of course, if the root indices are the same, yes, for example, a second degree here and a third degree there, and a third degree there. If they are different, then you

05:40

Speaker A

cannot do it. We bring them under a common root, multiply them, and maybe then it will be possible to find the root of that product. The nth root of a divided by b. Well, this is, in principle, expected, since if we have a

05:56

Speaker A

root of a product, then why wouldn't we have a root of a quotient? It also breaks down. You can find the root of the numerator and the root of the denominator separately and then divide them. This formula also works in the

06:10

Speaker A

reverse direction. The root of the degree of the root of degree k of number a can all be simplified. The root will be of degree n times k of number a. For example, you had the square root of the cube root of 64. 2\*3

06:36

Speaker A

= 6, the 6th root of 64. Well, and those who know mathematics well will easily give me the answer as to what number was hidden there. For the correct answer, you get a like from me, and generally, don't be shy about

06:52

Speaker A

writing comments. I will answer everyone. The more comments, the more interesting the communication. Let's continue. The n-th root of a is equal to the nk-th root of a to the power of k. It's very similar to the previous

07:11

Speaker A

point. If I have an inconvenient number, I can increase it by raising it to a power, but the root must also be increased accordingly. For example, we had two times the square root of three.

07:28

Speaker A

I can multiply the index by two, and it becomes the 6th root of nine or 3 squared. Different situations can arise. This is applied individually to each example. We will examine this now, and it will become clearer. The n-th root

07:44

Speaker A

of a to the k-th power equals the n-th root of a, all raised to the power of k. Sometimes it is more convenient not to raise to a power and then find the root, but to find the root first and

07:55

Speaker A

then raise it to a power. The nk-th root of a to the power of nk is equal to a to the power of k. That is, if I have a number raised to the same power as the root being taken, then it will

08:10

Speaker A

simply be a to the power of k. And, of course, the simplest case: the n-th root of a to the power of n is just the number a. The arithmetic n-th root of a number a, where a is greater than or

08:30

Speaker A

equal to zero, is a non-negative number whose n-th power equals a, and it is denoted as follows. The n-th root of a, where n is a natural number greater than one, is called the root index, and the number a itself is the radicand;

08:43

Speaker A

this is very similar to the previous definition, but only for positive numbers. In other words, there is this limitation. From the definition of an arithmetic root, it follows that: the expression for the n-th root of a is defined only if a is greater than or

09:00

Speaker A

equal to zero. The expression for the n-th root of a can only take on non-negative values. Since the radicand is greater than zero, the result must also be non-negative. And the equality of the n-th root of a to the power of n

09:17

Speaker A

equals a is correct for any non-negative value of a. Well, before we move on to solving examples and problems, I want to thank the sponsors of our channel. Here are our wonderful supporters. Ihor Klykach, Kateryna Yarotska, Palamarenko, supporters of

09:36

Speaker A

the Teacher OFM channel. Svitlana Danyliuk, Semen Petrov, Artem B12, Liubov Cherneha. All these people help the channel. Their avatars are on the main page of the channel. Accordingly, each of you can sponsor our channel.

09:55

Speaker A

Thank you to everyone who joins in and helps to create new video lessons. Now let's move on to the examples. Let's find the arithmetic roots of the numbers. The 4th root of 16. Well, what number must be raised to the fourth

10:11

Speaker A

power to get 16? Of course, it is the number two. The square root

10:27

Speaker A

two can be omitted. What number needs to be raised? Well, 0.9, and that will be 0.81. The cube root of 216 is the number 6. 6³ equals 216. One could write the cube root of 6³ under the radical, and then the cube root of 6³.

10:51

Speaker A

By property, they cancel out, leaving only six. Yes, the square root of 169. Well, 13 on top, 10 on the bottom. 13² is 169, 10² is 100. Nothing complicated. Let's simplify the expression: cube root of 8\*m³\*n⁹. So , I will break it down into three

11:14

Speaker A

factors, into three separate roots. I have every right to do this according to the properties of nth-degree roots.

11:22

Speaker A

I represented the eight as 2³. n⁹ could be represented as n³ raised to the power of two. What do we get then?

11:32

Speaker A

The cube root of 2³ equals two. Here it's just m. Here we have n to the third power, right? n³ raised to the third power. 3\*3 = 9. Next, a more complex expression: 3\*square root of ( a⁵\*x²)/18. So, I clearly see x²

11:57

Speaker A

first, I take it out, x will be on top, I will write one on the bottom so the fraction isn't lost. And for a⁵, I break it down into a⁴ multiplied by a . Well, here a to the first can be

12:09

Speaker A

written, 4 + 1 = 5. And 18 is broken down into 2\*9. And x cancels out, it cancels out. It will be 3/1. I pull out a⁴, it becomes a². I pulled out the nine, it becomes a three. Under the

12:30

Speaker A

root, a remains divided by 2; 3/3 cancels out, it will be a², and the root of a/2. That's it, I simplified the expression. Square root of 4, oh, square root of the number 4. That four confused me. multiplied by (3-√10)

12:52

Speaker A

squared. Once again, there is a multiplication here that is usually not written, but I can break it into factors and separately extract the square root of 4, which will be two, and the square root of (3-√10) squared will be 3-√10. Here is the

13:09

Speaker A

square root, here is the square. The same expression. 2\*3 is 6, 2 times √10 will be 2√10. That's it, there's nothing left to simplify here.

13:19

Speaker A

Number three. Let's find the value of the expression. Yes, cube root of 2\* cube root of 500. Hmm. There is no such number that when raised to the third power equals two. And there is no such number that when raised to the third

13:35

Speaker A

power equals 500. But if I bring them under a common radical, it will be the cube root of 2\*500, which is 1000. Aha.

13:43

Speaker A

And 10, if I multiply it by itself three times, will be exactly 1000. Therefore, the cube root of 1000 is the number 10. Cube root of -1 and 91/125.

13:57

Speaker A

Yes, it's a fraction here. We need to get rid of the whole part. To do this, I multiply the denominator by the whole part and add the numerator. 125\*1 + 91 will be 216 on top. 125 at the bottom.

14:14

Speaker A

The number is negative, but the root index is also an odd number, so the root can be extracted and it exists. It will be -6/5. One could write -6 cubed, 5 cubed, and they all cancel out, leaving -6/5. Cube root of the number

14:32

Speaker A

0.27\*3 ^ 6. Well, this is simpler. 0.27 can be represented as the number 0.3 raised to the third power. Extracting each separately, it means it will be 0.3\*3². 9\*0.3 will be 2.7. That's it, it seems I didn't make any mistakes. If

14:54

Speaker A

I did, those who watch my video lessons carefully should pause the video and write in the comments at what minute and second they heard or saw a mistake.

15:06

Speaker A

Whoever finds it correctly and indicates what the correct answer should be, the correct number, will get a like from me. So, be vigilant. Fourth root of the number 48\*27. Yes, well, breaking it down by analogy with the third example makes no sense here,

15:24

Speaker A

because there is no such number that would go to the fourth power to make 48 , let alone 27. But let me multiply 48 by 27. That will be 1296. Oh, and this will be 6 to the fourth power. I wrote

15:42

Speaker A

it twice. 6 to the fourth power and the fourth root will just be six. Just great. Yes, another one, number four.

15:54

Speaker A

Let's simplify the expression. Oh. 2√5-2√45 + 2√20. The radicals cannot be combined here because there is subtraction and addition. If there were multiplication, it would be simpler. But there is a two next to each of the numbers. A factor, right?

16:13

Speaker A

These are 2\*√5, 2\*√45, and 2\*√20. I am factoring this out. And I see that I can break down the number 45 into 5\*9 and 5\*4. Well, the number 20. I can do that. I can calmly do it, I can factor

16:35

Speaker A

it. We can either multiply or factor. Moving on. The two remains, √5 remains. √9 will be 3. We take it out from under the radical sign. √4 is 2.

16:50

Speaker A

We take it out from under the radical sign. Hmm. What do I see? Next to each one is √5. We factor out the common factor. It will be 2√5. Here, one remains. Here -3, here 2. 1-3 is -2. -2

17:05

Speaker A

+ 2, oops, zero. That is, 2√5\*0. So the entire expression equals zero. Cool . We solved it. We solved it just to get zero. Well, the main thing is that we solved it. Our task was to exercise our brains. Next example. Fourth root

17:25

Speaker A

of 3-fourth root of 27\*fourth root of 3 + fourth root of 27. So, does one have to multiply each one, right? Or maybe it's better to recall the formulas for abbreviated multiplication?(a-b)\*(a + b ), which equals a²-b². Only instead

17:48

Speaker A

of a, I have the 4th root of x, and instead of b, I have the 4th root of 27 . Let's use the formulas. Otherwise, why did we learn them earlier? Why? If we can't use them today, we definitely

18:02

Speaker A

should. It becomes the 4th root of 3 squared minus the 4th root of 27 squared. What do we have? Well, the square root of 3 and the square root of 27. And that's how everything simplifies according to the rules. It

18:22

Speaker A

will be a little bit easier. Yes, great . We’ve sorted that out. Homework. For those who want to get a like from me, you need to find the value of the expression: the 4th root of 324 multiplied by 4. Everyone who finds the

18:39

Speaker A

value, write the answer in the comments . If you calculated it correctly, you'll get a like from me. Agreed?

18:46

Speaker A

Great. Well, now we have a short music break, after which tests await us. So, don't change the channel. And so. We rested a little. Now, test assignments await us, onward for the grade. So, where are my tests? Here they are,

19:28

Speaker A

computer-based, no installation required. Just take them and use them. Nth root, arithmetic root of a degree, and its properties. Well, logical, yes, it matches the lesson topic. 13 questions, 15 minutes, 10th grade, algebra. Great. I am taking the tests,

19:44

Speaker A

entering my name, and starting the testing. And so, the nth root of the number zero equals zero, does not exist , or one. Before hearing the correct answer, pause the video and think about how you would answer. Compare with my

20:02

Speaker A

answer. If it matches, well done. Count how many answers matched and then write in the comments how many you got right.

20:12

Speaker A

Okay. I am curious to find out how well you understood this video. And so, here the answer is zero. The nth root of a divided by b equals. Meaning, this expression, this expression, or this expression. You need to pause the video

20:29

Speaker A

more often. Well, and I say the correct answer is the first one. Restore the definition by setting the correct sequence. The nth power of which equals a is called a number. n is an element of N. Yes, the nth root is not nonsense

20:44

Speaker A

yet. Well, that’s clear. Need to restore it. The nth root is called a number where n belongs to N, yes, n greater than one, of the number a, the nth power of which equals...aha. Like this: the nth root of the number a is

21:02

Speaker A

called a number whose nth power equals a. Now it's logical. Now I hope I restored it correctly. Power of a equals, meaning a equals this expression or this expression. Which is the correct option? Well, the second option is correct. The nth root of the

21:25

Speaker A

nth root of the number a is this option . Is it this option or this option? I think the second option is correct.

21:38

Speaker A

Establish the correspondence. So, this is the radical sign, the index of the root, and the radicand. Like this. Does a root of an even degree from a negative real number exist or not? An even root of a negative number does not

21:55

Speaker A

exist. Restore the definition by putting it in the correct order. The nth power of which equals a. A non-negative number of a number is called an arithmetic root. Yes, a non-negative number of a number is called the nth power, of which equals

22:13

Speaker A

yes, an arithmetic root of the nth degree of a number a is called a non-negative number, the nth power of which equals a. See, I restored it. A root of an even degree from a positive real number has an even degree positive

22:33

Speaker A

number. So, it has one value or two. I don't remember. If I make a mistake, the tests will show me later. Well, I think it's one. A root of an odd degree from any real number, yes, of an odd

23:03

Speaker A

degree also has one value. Yes, the root of the degree of a times b. What is the correct answer? It's this one: the nth root of a to the power of nk equals this answer, this answer. or this answer? well, something tells me

23:23

Speaker A

it's the first one...oh, yes, and this option: nth root. of a to the power of k equals: this option, this one, this one, or this one? something tells me it's this option. and let's look at the answer. so, I have one mistake, but I

23:42

Speaker A

got the maximum score. Meaning the system allows for one mistake. Of course, more mistakes mean a lower grade. An inverse relationship, there's no escaping that. But I can make a mistake once. So, let's look at the report. Where is that mistake? Of

24:00

Speaker A

course, one had to choose two values here. Explaining it to myself, right? There were two values. Remember, 3 to the 4th power is 81, -3 to the 4th power is 81. two. values. So you saw where you went wrong, analyzed it, and

24:15

Speaker A

you won't make such mistakes in the future. Here is the second reason why I made a mistake. What else can be done with the report? It can be saved to a file, it can be printed out. It shows

24:27

Speaker A

the lesson topic: who took it, when they took it, how long the test lasted, how many questions were asked, how many correct answers, and the grade. Well, yes. That's, in principle, all. Well, if you like my video lessons, be sure

24:42

Speaker A

to hit that arithmetic like button. It's not hard for you, but it's pleasant for me. Tell your friends about the video lesson and the Top School channel in real life and in virtual life. Subscribe to the channel, ring the bell so you don't miss new

24:58

Speaker A

video lessons. Become a sponsor of the channel. This will allow me to continue creating new video lessons with your support. Join us. We have a website where all video lessons are published.

25:12

Speaker A

A community on Facebook and Viber where we help each other with homework. Photos on Instagram, and short videos on TikTok. That's pretty much everything for today. Frames. Where are the frames? There should be frames. Oh, the frames are appearing as they should

25:33

Speaker A

. You know, three random comments get into a frame at the end of the lesson.

25:39

Speaker A

Well, I have a bonus task for you. For those who watched until the end, write the word autumn in capital letters in the comments. And I will know you watched to the end and will gladly hit the like button. That’s all for me

25:54

Speaker A

today. All the best and goodbye.

Topics: nth root arithmetic root algebra 10th grade math root properties even degree root odd degree root radicand exponentiation inverse math lesson

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