Let's try to figure out what kind of concept a "root" is and "what it is eaten with." To do this, consider examples that you have already encountered in the lessons (well, or you just have to face this).
For example, we have an equation. What is the solution to this equation? What numbers can be squared and get at the same time? Remembering the multiplication table, you can easily give the answer: and (because when you multiply two negative numbers, you get a positive number)! To simplify, mathematicians have introduced a special concept of the square root and assigned it a special symbol.
Let's define the arithmetic square root.
Why does the number have to be non-negative? For example, what is equal to. Okay, let's try to figure it out. Maybe three? Let's check: and not. Maybe, ? Again, check: Well, is it not selected? This is to be expected - because there are no numbers that, when squared, give a negative number!
This must be remembered: the number or expression under the root sign must be non-negative!
However, the most attentive have probably already noticed that the definition says that the solution of the square root of "a number is called such non-negative number whose square is ". Some of you will say that at the very beginning we analyzed the example, selected numbers that can be squared and obtained at the same time, the answer was and, and here it is talking about some kind of “non-negative number”! Such a remark is quite appropriate. Here it is necessary simply to distinguish between the concepts of quadratic equations and the arithmetic square root of a number. For example, it is not equivalent to an expression.
It follows that, that is, or. (Read the topic "")
And it follows that.
Of course, this is very confusing, but it must be remembered that the signs are the result of solving the equation, since when solving the equation, we must write down all the x's that, when substituted into the original equation, will give the correct result. In our quadratic equation fits both and.
However, if just take the square root from something, then always we get one non-negative result.
Now try to solve this equation. Everything is not so simple and smooth, right? Try to sort through the numbers, maybe something will burn out? Let's start from the very beginning - from scratch: - does not fit, move on - less than three, also brush aside, but what if. Let's check: - also does not fit, because it's more than three. With negative numbers, the same story will turn out. And what to do now? Did the search give us nothing? Not at all, now we know for sure that the answer will be some number between and, as well as between and. Also, it is obvious that the solutions will not be integers. Moreover, they are not rational. So, what is next? Let's build a graph of the function and mark the solutions on it.
Let's try to trick the system and get an answer with a calculator! Let's get the root out of business! Oh-oh-oh, it turns out that. This number never ends. How can you remember this, because there will be no calculator on the exam !? Everything is very simple, you don’t need to remember it, you need to remember (or be able to quickly estimate) an approximate value. and the answers themselves. Such numbers are called irrational, and it was to simplify the notation of such numbers that the concept of a square root was introduced.
Let's look at another example to reinforce. Let's analyze the following problem: you need to cross diagonally a square field with a side of km, how many km do you have to go?
The most obvious thing here is to consider the triangle separately and use the Pythagorean theorem:. Thus, . So what is the required distance here? Obviously, the distance cannot be negative, we get that. The root of two is approximately equal, but, as we noted earlier, is already a complete answer.
So that solving examples with roots does not cause problems, you need to see and recognize them. To do this, you need to know at least the squares of numbers from to, as well as be able to recognize them. For example, you need to know what is squared, and also, conversely, what is squared.
Did you figure out what a square root is? Then solve some examples.
Examples.
Well, how did it work? Now let's see these examples:
Answers:
cube root
Well, we sort of figured out the concept of a square root, now we will try to figure out what a cube root is and what is their difference.
The cube root of some number is the number whose cube is equal to. Have you noticed how much easier it is? There are no restrictions on the possible values of both the value under the cube root sign and the number to be extracted. That is, the cube root can be taken from any number:.
Caught what a cube root is and how to extract it? Then go ahead with examples.
Examples.
Answers:
Root - oh degree
Well, we figured out the concepts of square and cube roots. Now we generalize the obtained knowledge by the concept th root.
th root from a number is a number whose th power is equal, i.e.
is tantamount to.
If - even, then:
- with negative, the expression does not make sense (the roots of an even -th degree of negative numbers cannot be extracted!);
- with non-negative() expression has one non-negative root.
If - is odd, then the expression has a single root for any.
Do not be alarmed, the same principles apply here as with square and cube roots. That is, the principles that we applied when considering square roots are extended to all roots of an even -th degree.
And those properties that were used for the cube root apply to the roots of an odd th degree.
Well, it became clearer? Let's understand with examples:
Here everything is more or less clear: first we look - yeah, the degree is even, the number under the root is positive, so our task is to find a number whose fourth degree will give us. Well, any guesses? Maybe, ? Exactly!
So, the degree is equal - odd, under the root the number is negative. Our task is to find such a number, which, when raised to a power, turns out. It is quite difficult to immediately notice the root. However, you can narrow down your search right away, right? Firstly, the desired number is definitely negative, and secondly, it can be seen that it is odd, and therefore the desired number is odd. Try to pick up the root. Of course, and you can safely brush aside. Maybe, ?
Yes, this is what we were looking for! Note that to simplify the calculation, we used the properties of degrees: .
Basic properties of roots
Understandably? If not, then after considering the examples, everything should fall into place.
Root multiplication
How to multiply roots? The simplest and most basic property helps answer this question:
Let's start with a simple one:
The roots of the resulting numbers are not exactly extracted? Don't worry, here are some examples:
But what if there are not two multipliers, but more? The same! The root multiplication formula works with any number of factors:
What can we do with it? Well, of course, hide the triple under the root, while remembering that the triple is the square root of!
Why do we need it? Yes, just to expand our capabilities when solving examples:
How do you like this property of roots? Makes life much easier? For me, that's right! You just have to remember that we can only add positive numbers under the sign of the root of an even degree.
Let's see where else it can come in handy. For example, in a task you need to compare two numbers:
That more:
You won't say right off the bat. Well, let's use the parsed property of adding a number under the root sign? Then forward:
Well, knowing that the larger the number under the sign of the root, the larger the root itself! Those. if means . From this we firmly conclude that And no one will convince us otherwise!
Before that, we introduced a factor under the sign of the root, but how to take it out? You just need to factor it out and extract what is extracted!
It was possible to go the other way and decompose into other factors:
Not bad, right? Any of these approaches is correct, decide how you feel comfortable.
For example, here's an expression:
In this example, the degree is even, but what if it is odd? Again, apply the power properties and factor everything:
Everything seems to be clear with this, but how to extract a root from a number in a degree? Here, for example, is this:
Pretty simple, right? What if the degree is greater than two? We follow the same logic using the properties of degrees:
Well, is everything clear? Then here's an example:
These are pitfalls, about them always worth remembering. This is actually a reflection on the property examples:
| for odd: for even and: |
Understandably? Fix it with examples:
Yeah, we see the root to an even degree, the negative number under the root is also to an even degree. Well, does it work the same? And here's what:
That's all! Now here are some examples:
Got it? Then go ahead with examples.
Examples.
Answers.
If you received answers, then you can move on with peace of mind. If not, then let's look at these examples:
Let's look at two other properties of roots:
These properties must be analyzed in examples. Well, shall we do this?
Got it? Let's fix it.
Examples.
Answers.
ROOTS AND THEIR PROPERTIES. MIDDLE LEVEL
Arithmetic square root
The equation has two solutions: and. These are numbers whose square is equal.
Consider the equation. Let's solve it graphically. Let's draw a graph of the function and a line on the level. The points of intersection of these lines will be the solutions. We see that this equation also has two solutions - one positive, the other negative:
But in this case, the solutions are not integers. Moreover, they are not rational. In order to write down these irrational decisions, we introduce a special square root symbol.
Arithmetic square root is a non-negative number whose square is . When the expression is not defined, because there is no such number, the square of which is equal to a negative number.
Square root: .
For example, . And it follows that or.
Again, this is very important: The square root is always a non-negative number: !
cube root out of number is the number whose cube is equal. The cube root is defined for everyone. It can be extracted from any number: . As you can see, it can also take negative values.
The root of the th degree of a number is the number whose th degree is equal to, i.e.
If - even, then:
- if, then the th root of a is not defined.
- if, then the non-negative root of the equation is called the arithmetic root of the th degree of and is denoted.
If - is odd, then the equation has a single root for any.
Have you noticed that we write its degree to the top left of the root sign? But not for the square root! If you see a root without a degree, then it is square (degrees).
Examples.
Basic properties of roots
ROOTS AND THEIR PROPERTIES. BRIEFLY ABOUT THE MAIN
Square root (arithmetic square root) from a non-negative number is called such non-negative number whose square is
Root properties:
Well, the topic is over. If you are reading these lines, then you are very cool.
Because only 5% of people are able to master something on their own. And if you have read to the end, then you are in the 5%!
Now the most important thing.
You've figured out the theory on this topic. And, I repeat, it's ... it's just super! You are already better than the vast majority of your peers.
The problem is that this may not be enough ...
For what?
For the successful passing of the exam, for admission to the institute on the budget and, MOST IMPORTANTLY, for life.
I will not convince you of anything, I will just say one thing ...
People who have received a good education earn much more than those who have not received it. This is statistics.
But this is not the main thing.
The main thing is that they are MORE HAPPY (there are such studies). Perhaps because much more opportunities open up before them and life becomes brighter? Don't know...
But think for yourself...
What does it take to be sure to be better than others on the exam and be ultimately ... happier?
FILL YOUR HAND, SOLVING PROBLEMS ON THIS TOPIC.
On the exam, you will not be asked theory.
You will need solve problems on time.
And, if you haven’t solved them (LOTS!), you will definitely make a stupid mistake somewhere or simply won’t make it in time.
It's like in sports - you need to repeat many times to win for sure.
Find a collection anywhere you want necessarily with solutions, detailed analysis and decide, decide, decide!
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“Understood” and “I know how to solve” are completely different skills. You need both.
Find problems and solve!
Lesson Objectives:
educational: create conditions for the formation of a holistic view of the root of the n-th degree, the skills of conscious and rational use of the properties of the root in solving various problems.
Educational: create conditions for the development of algorithmic, creative thinking, develop self-control skills.
Educational: to promote the development of interest in the subject, activity, to cultivate accuracy in work, the ability to express one's own opinion, to give recommendations.
During the classes
1. Organizational moment.
Good afternoon! Good hour!
How glad I am to see you.
The bell has already rung
The lesson starts.
They smiled. Leveled up.
looked at each other
And they sat down quietly.
2. Lesson motivation.
An outstanding French philosopher, scientist Blaise Pascal stated: "The greatness of man is in his ability to think." Today we will try to feel like great people by discovering knowledge for ourselves. The motto for today's lesson will be the words of the ancient Greek mathematician Thales:
What is the most in the world? - Space.
What is the fastest? - Mind.
What is the wisest? - Time.
What is the most enjoyable? - Achieve what you want.
I want each of you to achieve the desired result in today's lesson.
3. Actualization of knowledge.
1. Name mutually inverse algebraic operations on numbers. (Addition and subtraction, multiplication and division)
2. Is it always possible to perform such an algebraic operation as division? (No, you can't divide by zero)
3. What other operation can you perform with numbers? (Exponentiation)
4. What operation will be her reverse? (root extraction)
5. What degree root can you extract? (Second root)
6. What properties of the square root do you know? (Extracting the square root from a product, from a quotient, from a root, exponentiation)
7. Find the values of expressions:
From the history. Even 4000 years ago, Babylonian scientists compiled, along with multiplication tables and tables of reciprocals (with the help of which the division of numbers was reduced to multiplication), tables of squares of numbers and square roots of numbers. At the same time, they were able to find the approximate value of the square root of any integer.
4. Learning new material.
Obviously, in accordance with the basic properties of degrees with natural exponents, from any positive number there are two opposite values of the root of an even degree, for example, the numbers 4 and -4 are square roots of 16, since (-4) 2 \u003d 42 \u003d 16, and the numbers 3 and -3 are the fourth roots of 81, since (-3) 4 \u003d Z4 \u003d 81.
Also, there is no even root of a negative number, because an even power of any real number is non-negative. As for the root of an odd degree, then for any real number there is only one root of an odd degree from this number. For example, 3 is the third root of 27 because Z3 = 27, and -2 is the fifth root of -32 because (-2)5 = 32.
In connection with the existence of two roots of an even degree from a positive number, we introduce the concept of an arithmetic root in order to eliminate this ambiguity of the root.
A non-negative value of the n-th root of a non-negative number is called an arithmetic root.
Designation: - the root of the n-th degree.
The number n is called the degree of the arithmetic root. If n = 2, then the degree of the root is not indicated and is written. The root of the second degree is called the square root, and the root of the third degree is called the cubic root.
B, b2 = a, a ≥ 0, b ≥ 0
B, bp = a, p - even a ≥ 0, b ≥ 0
p - odd a, b - any
Properties
1. , a ≥ 0, b ≥ 0
2. , a ≥ 0, b > 0
3. , a ≥ 0
4. , m, n, k - natural numbers
5. Consolidation of new material.
oral work
a) What expressions make sense?
b) For what values of the variable a does the expression make sense?
Solve #3, 4, 7, 9, 11.
6. Physical education.
In all matters, moderation is needed,
Let it be the main rule.
Do gymnastics, if you thought for a long time,
Gymnastics does not exhaust the body,
But it cleanses the whole body!
Close your eyes, relax your body
Imagine - you are birds, you suddenly flew!
Now you swim like a dolphin in the ocean,
Now in the garden you pick ripe apples.
Left, right, looked around
Open your eyes and get back to work!
7. Independent work.
Working in pairs with 178 #1, #2.
8. D / z. Learn item 10 (p.160-161), solve No. 5, 6, 8, 12, 16 (1, 2).
9. The results of the lesson. Reflection of activity.
Did the lesson achieve its purpose?
What have you learned?
Video lesson 2: Root properties of degree n > 1
Lecture: Root of degree n > 1 and its properties
Root
Suppose you have an equation like:
The solution to this equation will be x 1 \u003d 2 and x 2 \u003d (-2). Both solutions are suitable as an answer, since numbers with equal modules, when raised to an even power, give the same result.
This was a simple example, however, what can we do if, for example,
Let's try to graph the function y=x 2 . Its graph is a parabola:
On the graph, you need to find points that correspond to the value y \u003d 3. These points are:
This means that this value cannot be called an integer, but can be represented as a square root.
Any root is irrational number. Irrational numbers include roots, non-periodic infinite fractions.
Square root is a non-negative number "a", the radical expression of which is equal to the given number "a" squared.
For example,
That is, as a result, we will get only a positive value. However, as a solution to a quadratic equation of the form
The solution will be x 1 = 4, x 2 = (-4).
Square root properties
1. Whatever value x takes, this expression is true in any case:
2. Comparison of numbers containing a square root. To compare these numbers, it is necessary to enter both one and the second number under the root sign. That number will be greater whose radical expression is greater.
We enter the number 2 under the sign of the root
Now let's put the number 4 under the root sign. As a result of this, we get
And only now the two resulting expressions can be compared:
3. Removing the multiplier from under the root.
If the radical expression can be decomposed into two factors, one of which can be taken out of the subsign of the root, then this rule must be used.
4. There is a property inverse to this - introducing a multiplier under the root. We obviously used this property in the second property.
Rootn-th degree and its properties
What is a rootnth degree? How to extract the root?
In the eighth grade, you already managed to get acquainted with square root. We solved typical examples with roots, using certain properties of the roots. Also decided quadratic equations, where without extracting the square root - no way. But the square root is just a special case of a broader concept - root n th degree . In addition to the square, there is, for example, a cube root, a root of the fourth, fifth and higher degrees. And for successful work with such roots, it would still be nice to start with “you” with square roots.) Therefore, for those who have problems with them, I strongly recommend repeating.
Extracting a root is one of the inverse operations of exponentiation.) Why "one of"? Because, extracting the root, we are looking for base according to famous degree and indicator. And there is another inverse operation - finding indicator according to famous degree and basis. This operation is called finding logarithm. It is more complex than extracting the root and is studied in high school.)
So let's get acquainted!
First, the designation. The square root, as we already know, is denoted like this:. This icon is called very beautifully and scientifically - radical. And what are the roots of other degrees? It's very simple: above the "tail" of the radical, they additionally write an indicator of the degree whose root is being sought. If you are looking for a cube root, then write a triple: . If the root of the fourth degree, then, respectively, . And so on.) In general, the root of the nth degree is denoted like this:
Where .
Numbera , as in square roots, is called radical expression and here is the numbern this is new for us. And called root indicator .
How to extract roots of any degrees? Just like square ones - figure out what number to the nth power gives us a numbera .)
How, for example, to extract the cube root of 8? I.e ? And what number cubed will give us 8? Deuce, of course.) So they write:
Or . What is the number to the fourth power of 81? Three.) So,
What about the tenth root of 1? Well, it’s a no brainer that a unit to any power (including the tenth) is equal to one.) That is:
And generally speaking .
With zero, the same story: zero to any natural power is equal to zero. That is, .
As you can see, in comparison with square roots, it is already more difficult to figure out which number gives us the root number to one degree or anothera . More difficult pick up answer and check it for correctness by exponentiationn . The situation is greatly facilitated if you know in person the degree of popular numbers. So now we are training. :) We recognize the degrees!)
Answers (in disarray):
Yes Yes! There are more answers than tasks.) Because, for example, 2 8 , 4 4 and 16 2 are all the same number 256.
Trained? Then we consider examples:
Answers (also in disarray): 6; 2; 3; 2; 3; 5.
Happened? Fabulous! Let's move on.)
Root restrictions. arithmetic rootnth degree.
In the roots of the nth degree, as well as in the square, there are also limitations and their chips. At their core, they are no different from those restrictions for square roots.
Doesn't get picked, does it? What is 3, what is -3 to the fourth power will be +81. :) And with any root even degree from a negative number will be the same song. And this means that it is impossible to extract even roots from negative numbers . This is a forbidden action in mathematics. As forbidden as dividing by zero. Therefore, expressions such as , and the like - don't make sense.
But the roots odd degrees of negative numbers - please!
For example, ; 
, etc.)
And from positive numbers, you can safely extract any roots, any degrees:
In general, it's understandable, I think.) And, by the way, the root does not have to be extracted exactly. These are just examples, purely for understanding.) It happens that in the process of solving (for example, equations) rather bad roots come up. Something like . From the eight, the cube root is extracted perfectly, and here the seven is under the root. What to do? It's OK. Everything is exactly the same.- this is the number that, when cubed, will give us 7. Only the number is very ugly and shaggy. Here it is:
Moreover, this number never ends and has no period: the numbers follow completely randomly. It is irrational ... In such cases, the answer is left in the form of a root.) But if the root is extracted purely (for example,), then, naturally, the root must be calculated and written down:
Again we take our experimental number 81 and extract the fourth root from it:
Because three in the fourth will be 81. Well, good! But also minus three the fourth will also be 81!
There is an ambiguity:
And, in order to eliminate it, just as in square roots, a special term was introduced: arithmetic rootnth degree from among a - it's like that non-negative number,n-th degree of which is equal to a .
And the answer with plus or minus is called differently - algebraic rootnth degree. For any even power, the algebraic root will be two opposite numbers. At school, they work only with arithmetic roots. Therefore, negative numbers in arithmetic roots are simply discarded. For example, they write: The plus itself, of course, is not written: it imply.
Everything, it would seem, is simple, but ... But what about the roots of an odd degree from negative numbers? After all, there is always a negative number when extracting! Since any negative number in odd degree also gives a negative number. And the arithmetic root only works with non-negative numbers! That's why it's arithmetic.)
In such roots, they do this: they take out a minus from under the root and put it in front of the root. Like this:
In such cases it is said that expressed in terms of an arithmetic (i.e. already non-negative) root .
But there is one thing that can be confusing - this is the solution of simple equations with powers. For example, here's an equation:
We write the answer: In fact, this answer is just an abbreviated notation two answers:
The misunderstanding here is that I already wrote a little higher that only non-negative (i.e. arithmetic) roots are considered at school. And here is one of the answers with a minus ... How to be? No way! The signs here are the result of solving the equation. BUT the root itself- the value is still non-negative! See for yourself:
Well, is it clearer now? with brackets?)
With an odd degree, everything is much simpler - it always turns out one root. Plus or minus. For example:
So if we simply we extract the root (of an even degree) from the number, then we always get one non-negative result. Because it is an arithmetic root. Now, if we decide the equation with an even degree, we get two opposite roots, since this is solution of the equation.
With roots of odd degrees (cubic, fifth degree, etc.) there are no problems. We extract ourselves and do not bathe with signs. Plus under the root means the result of extraction with a plus. Minus means minus.
And now it's time to meet root properties. Some will already be familiar to us from square roots, but a few new ones will be added. Go!
Root properties. The root of the work.
This property is already familiar to us from square roots. For roots of other degrees, everything is similar:
I.e, the root of the product is equal to the product of the roots of each factor separately.
If the indicatorn even, then both radical numbersa andb must, of course, be non-negative, otherwise the formula has no meaning. In the case of an odd indicator, there are no restrictions: we take the minuses forward from under the roots and then work with arithmetic roots.)
As in square roots, here this formula is equally useful both from left to right and from right to left. Applying the formula from left to right allows you to extract the roots from the work. For example:
This formula, by the way, is valid not only for two, but for any number of factors. For example:
Also, using this formula, you can extract roots from large numbers: for this, the number under the root is decomposed into smaller factors, and then the roots are extracted separately from each factor.
For example, such a task:
The number is large enough. Does it take root? smooth- also without a calculator it is not clear. It would be nice to factor it out. What exactly is the number 3375 divisible by? By 5, it seems: the last digit is five.) Divide:
Oh, divisible by 5 again! 675:5 = 135. And 135 is again divided by five. Yes, when will it end?
135:5 = 27. With the number 27, everything is already clear - this is a three in a cube. Means,
Then:
They took the root piece by piece, well, okay.)
Or this example:
Again, we factorize according to the signs of divisibility. What? On 4, because the last pair of numbers 40 is divisible by 4. And by 10, because the last digit is zero. So, you can divide in one fell swoop by 40 at once:
About the number 216, we already know that this is a six cubed. That is,
And 40, in turn, can be decomposed as . Then
And then finally we get:
It didn’t work out cleanly to extract the root, well, that’s okay. Anyway, we have simplified the expression: we know that it is customary to leave the smallest possible number under the root (even if square, even if cubic - any). In this example, we performed one very useful operation, also already familiar to us from square roots. Do you recognize? Yes! We endured factors from under the root. In this example, we took out a deuce and a six, i.e. number 12.
How to take the factor out of the sign of the root?
It is very easy to take out the factor (or factors) beyond the root sign. We decompose the root expression into factors and extract what is extracted.) And what is not extracted, we leave it at the root. See:
We decompose the number 9072 into factors. Since we have a root of the fourth degree, first of all we try to decompose into factors that are the fourth powers of natural numbers - 16, 81, etc.
Let's try to divide 9072 by 16:
Shared!
But 567 seems to be divisible by 81:
Means, .
Then
Root properties. Root multiplication.
Consider now the reverse application of the formula - from right to left:
At first glance, nothing new, but appearances are deceiving.) Reverse application of the formula greatly expands our capabilities. For example:
Hmm, so what's wrong with that? They multiplied everything. There really is nothing special here. The usual multiplication of roots. And here's an example!
Separately, the roots are not purely extracted from the factors. But the result is excellent.)
Again, the formula is valid for any number of factors. For example, you need to calculate the following expression:
The main thing here is attention. The example contains various roots are cubic and fourth degree. And none of them are definitely extracted ...
And the formula for the product of roots is applicable only to roots with the same indicators. Therefore, we group the cube roots into a separate pile and into a separate pile - the fourth degree. And there, you see, everything will grow together.))
And I didn't need a calculator.
How to add a multiplier under the root sign?
The next useful thing is entering a number under the root. For example:
Is it possible to remove the triple inside the root? Elementary! If the triple is turned into root, then the formula for the product of the roots will work. So, we turn the three into a root. Since we have a root of the fourth degree, then we will also turn it into a root of the fourth degree.) Like this:
Then
The root, by the way, can be made from any non-negative number. Moreover, to the extent that we want (everything depends on a specific example). This will be the root of the nth power of this very number:
And now - Attention! Source of very gross errors! I didn't say anything here for nothing non-negative numbers. The arithmetic root works only with such. If we have a negative number somewhere in the task, then we either leave the minus in front of the root (if it is outside), or get rid of the minus under the root, if it is inside. I remind you if under the root even degree turns out to be a negative number, then the expression doesn't make sense.
For example, such a task. Enter a multiplier under the root sign:
If we now root minus two, then we will be cruelly mistaken:
What is wrong here? And the fact that the fourth degree, due to its parity, safely “ate” this minus, as a result of which a deliberately negative number turned into a positive one. The correct solution looks like this:
In the roots of odd degrees, the minus, although not “eaten”, is also better to leave it outside:
Here the root of an odd degree is cubic, and we have every right to drive the minus under the root too. But it is preferable in such examples to also leave the minus outside and write the answer expressed through the arithmetic (non-negative) root, since the root, although it has the right to life, but is not arithmetic.
So, with the introduction of a number under the root, everything is also clear, I hope.) Let's move on to the next property.
Root properties. The root of the fraction. Division of roots.
This property also completely repeats that for square roots. Only now we extend it to roots of any degree:
The root of a fraction is the root of the numerator divided by the root of the denominator.
If n is even, then the numbera must be non-negative, and the numberb - strictly positive (you cannot divide by zero). In the case of an odd exponent, the only constraint will be .
This property allows you to easily and quickly extract roots from fractions:
The idea is clear, I think. Instead of working with the fraction as a whole, we move on to working separately with the numerator and separately with the denominator.) If the fraction is a decimal or, horror, a mixed number, then we first proceed to ordinary fractions:
Now let's see how this formula works from right to left. Here, too, very useful possibilities are revealed. For example, this example:
The roots are not exactly extracted from the numerator and denominator, but from the whole fraction it’s fine.) You can solve this example in a different way - take out the factor in the numerator from under the root, followed by reduction:
As you wish. The answer is always the same - the correct one. If you don't make mistakes along the way.)
So, we figured out the multiplication / division of the roots. We rise to the next step and consider the third property - root to degree and root of degree .
Root to degree. Root of degree.
How to raise a root to a power? For example, let's say we have a number . Can this number be raised to a power? In a cube, for example? Certainly! Multiply the root by itself three times, and - according to the formula for the product of the roots:
Here is the root and degree as if mutually canceled out or compensated. Indeed, if we raise a number that, when cubed, will give us a triple, we raise it to this very cube, then what will we get? Three and get, of course! And so it will be for any non-negative number. In general:
If the exponents and the root are different, then there is no problem either. If you know the properties of degrees.)
If the exponent is less than the exponent of the root, then we simply drive the exponent under the root:
In general it will be:
The idea is clear: we raise the radical expression to a power, and then we simplify it by taking factors out from under the root, if possible. If an straight, thena must be non-negative. Why is understandable, I think.) And ifn odd, then no restrictions ona already gone:
Let's deal with now root of degree . That is, not the root itself will be raised to a power, but radical expression. There is nothing complicated here either, but there is much more scope for errors. Why? Because negative numbers come into play, which can confuse the signs. For now, let's start with the roots of odd powers - they are much simpler.
Let's say we have the number 2. Can we cube it? Certainly!
And now - back extract the cube root from the eight:
They started with a deuce, and returned to a deuce.) No wonder: raising to a cube was compensated by the inverse operation - extracting the cube root.
Another example:
Here, too, everything is on track. The degree and the root of each other compensated. In general, for the roots of odd degrees, we can write the following formula:
This formula is valid for any real numbera . Whether positive or negative.
That is, an odd degree and a root of the same degree always compensate each other and a radical expression is obtained. :)
But with even degree, this focus may no longer pass. See for yourself:
There is nothing special here yet. The fourth degree and the root of the fourth degree also balanced each other and it turned out just a deuce, i.e. rooted expression. And for anyone non-negative numbers will be the same. And now we just replace two in this root with minus two. So let's take a root like this:
The minus of the deuce safely “burned out” due to the fourth degree. And as a result of extracting the root (arithmetic!) We got positive number. It was minus two, it became plus two.) But if we just thoughtlessly “reduced” the degree and root (the same!), We would get
Which is the biggest mistake, yes.
Therefore, for even The formula for the root of the exponent looks like this:
Here, the module sign, unloved by many, was added, but there is nothing terrible in it: thanks to it, the formula also works for any real numbera. And the module simply cuts off the cons:
Only in the roots of the nth degree did an additional distinction appear between even and odd degrees. Even degrees, as we see, are more capricious, yes.)
And now consider a new useful and very interesting property, already characteristic of the roots of the nth degree: if the root exponent and the exponent of the root expression are multiplied (divided) by the same natural number, then the value of the root will not change.
Something reminiscent of the basic property of a fraction, isn't it? In fractions, we can also multiply (divide) the numerator and denominator by the same number (except zero). In fact, this property of the roots is also a consequence of the basic property of the fraction. When we get to know degree with a rational exponent then everything will become clear. What, how and where.)
The direct application of this formula allows us to simplify absolutely any roots from any degrees. Including, if the exponents of the root expression and the root itself various. For example, let's simplify the following expression:
We act simply. For starters, we single out the fourth degree from the tenth under the root and - go ahead! How? By the properties of degrees, of course! We take out the factor from under the root or work according to the formula of the root from the degree.
But let's simplify, using just this property. To do this, we represent the four under the root as:
And now - the most interesting - we reduce mentally the indicator under the root (two) with the root indicator (four)! And we get:
- The arithmetic root of a natural degree n>=2 from a non-negative number a is some non-negative number, when raised to the power of n, the number a is obtained.
It can be proved that for any non-negative a and natural n, the equation x^n=a will have one single non-negative root. It is this root that is called the arithmetic root of the nth degree from the number a.
The arithmetic root of the nth degree from the number a is denoted as follows n√a. The number a in this case is called the root expression.
The arithmetic root of the second degree is called the square root, and the arithmetic root of the third degree is called the cube root.
Basic properties of the arithmetic root of the nth degree
- 1. (n√a)^n = a.
For example, (5√2)^5 = 2.
This property follows directly from the definition of the arithmetic root of the nth degree.
If a is greater than or equal to zero, b is greater than zero, and n, m are some natural numbers such that n is greater than or equal to 2 and m is greater than or equal to 2, then the following properties are true:
- 2. n√(a*b)= n√a*n√b.
For example, 4√27 * 4√3 = 4√(27*3) = 4√81 =4√(3^4) = 3.
- 3. n√(a/b) = (n√a)/(n√b).
For example, 3√(256/625) :3√(4/5) = 3√((256/625) : (4/5)) = (3√(64))/(3√(125)) = 4/5.
- 4. (n√a)^m = n√(a^m).
For example, 7√(5^21) = 7√((5^7)^3)) = (7√(5^7))^3 = 5^3 = 125.
- 5. m√(n√a) = (n*m) √a.
For example, 3√(4√4096) = 12√4096 = 12√(2^12) = 2.
Note that in property 2, the number b can be equal to zero, and in property 4, the number m can be any integer, provided that a>0.
Proof of the second property
All the last four properties are proved similarly, so we restrict ourselves to proving only the second one: n√(a*b)= n√a*n√b.
Using the definition of an arithmetic root, we prove that n√(a*b)= n√a*n√b.
To do this, we prove two facts that n√a*n√b. Greater than or equal to zero, and that (n√a*n√b.)^n = ab.
- 1. n√a*n√b is greater than or equal to zero, since both a and b are greater than or equal to zero.
- 2. (n√a*n√b)^n = a*b since (n√a*n√b)^n = (n√a)^n *(n√b)^n = a* b.
Q.E.D. So the property is true. These properties will very often have to be used when simplifying expressions containing arithmetic roots.
