Topic "Root of a degree" n"It is advisable to split it into two lessons. In the first lesson, consider the cube root, compare its properties with the arithmetic square root and consider the graph of this Cube root function. Then in the second lesson, students will better understand the concept of a crown n-th degree. Comparing the two types of roots will help you avoid “typical” errors in the presence of values from negative expressions under the root sign.
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"Cubic root"
Lesson topic: Cube root
Zhikharev Sergey Alekseevich, mathematics teacher, MKOU “Pozhilinskaya Secondary School No. 13”
Lesson objectives:
- introduce the concept of cube root;
- develop skills in calculating cube roots;
- repeat and generalize knowledge about the arithmetic square root;
- continue preparing for the State Examination.
Checking the d.z.
One of the numbers below is marked on the coordinate line with a dot A. Enter this number.
What concept are the last three tasks related to?
What is the square root of a number? A ?
What is the arithmetic square root of a number? A ?
What values can it take? square root?
Can a radical expression be a negative number?
Among these geometric bodies, name a cube
What properties does a cube have?
How to find the volume of a cube?
Find the volume of a cube if its sides are equal:
Let's solve the problem
The volume of the cube is 125 cm³. Find the side of the cube.
Let the edge of the cube be X cm, then the volume of the cube is X³ cm³. By condition X³ = 125.
Hence, X= 5 cm.
Number X= 5 is the root of the equation X³ = 125. This number is called cube root or third root from number 125.
Definition.
The third root of the number A this number is called b, the third power of which is equal to A .
Designation.
Another approach to introducing the concept of cube root
For a given cubic function value A, you can find the value of the argument of the cubic function at this point. It will be equal, since extracting the root is the inverse action of raising to a power.
Square roots.
Definition. The square root of a name the number whose square is equal to A .
Definition. Arithmetic square root of a is a non-negative number whose square is equal to A .
Use the designation:
At A
Cube roots.
Definition. cube root from number a name the number whose cube is equal to A .
Use the designation:
"The cube root of A", or
"The 3rd root of A »
The expression makes sense for any A .
Launch the MyTestStudent program.
Open the “9th grade lesson” test.
A minute of rest
In what lessons or
you met in life
with the concept of root?
"Equation"
When you solve an equation, my friend,
You must find him spine.
The meaning of a letter is easy to check,
Put it into the equation carefully.
If you achieve true equality,
That root call the meaning immediately.
How do you understand Kozma Prutkov’s statement “Look to the root.”
When is this expression used?
In literature and philosophy there is the concept of “The Root of Evil.”
How do you understand this expression?
In what sense is this expression used?
Think about it, is it always easy and accurate to extract the cube root?
How can you find approximate cube root values?
Using the graph of a function at = X³, you can approximately calculate the cube roots of some numbers.
Using the graph of a function
at = X³ orally find the approximate meaning of the roots.
Do functions belong to the graph?
dots: A(8;2); In (216;–6)?
Can the radical expression of a cube root be negative?
What is the difference between a cube root and a square root?
Can the cube root be negative?
Define a root of the third degree.
The basic properties of the power function are given, including formulas and properties of the roots. Derivative, integral, expansion in power series and representation through complex numbers of a power function.
Definition
Definition
Power function with exponent p is the function f (x) = x p, the value of which at point x is equal to the value of the exponential function with base x at point p.
In addition, f (0) = 0 p = 0 for p > 0
.
For natural values exponent, the power function is the product of n numbers equal to x:
.
It is defined for all valid .
For positive rational values of the exponent, the power function is the product of n roots of degree m of the number x:
.
For odd m, it is defined for all real x. For even m, the power function is defined for non-negative ones.
For negative , the power function is determined by the formula:
.
Therefore, it is not defined at the point.
For irrational values of the exponent p, the power function is determined by the formula:
,
where a is an arbitrary positive number, not equal to one: .
When , it is defined for .
When , the power function is defined for .
Continuity. A power function is continuous in its domain of definition.
Properties and formulas of power functions for x ≥ 0
Here we will consider the properties of the power function for non-negative values of the argument x. As stated above, for some values of the exponent p, the power function is also defined for negative values of x. In this case, its properties can be obtained from the properties of , using even or odd. These cases are discussed and illustrated in detail on the page "".
A power function, y = x p, with exponent p has the following properties:
(1.1)
defined and continuous on the set
at ,
at ;
(1.2)
has many meanings
at ,
at ;
(1.3)
strictly increases with ,
strictly decreases as ;
(1.4)
at ;
at ;
(1.5)
;
(1.5*)
;
(1.6)
;
(1.7)
;
(1.7*)
;
(1.8)
;
(1.9)
.
Proof of properties is given on the page “Power function (proof of continuity and properties)”
Roots - definition, formulas, properties
Definition
Root of a number x of degree n is the number that when raised to the power n gives x:
.
Here n = 2, 3, 4, ...
- natural number, greater than one.
You can also say that the root of a number x of degree n is the root (i.e. solution) of the equation
.
Note that the function is the inverse of the function.
Square root of x is a root of degree 2: .
Cube root of x is a root of degree 3: .
Even degree
For even powers n = 2 m, the root is defined for x ≥ 0
. A formula that is often used is valid for both positive and negative x:
.
For square root:
.
The order in which the operations are performed is important here - that is, first the square is performed, resulting in a non-negative number, and then the root is taken from it (the square root can be taken from a non-negative number). If we changed the order: , then for negative x the root would be undefined, and with it the entire expression would be undefined.
Odd degree
For odd powers, the root is defined for all x:
;
.
Properties and formulas of roots
The root of x is a power function:
.
When x ≥ 0
the following formulas apply:
;
;
,
;
.
These formulas can also be applied for negative values of variables. You just need to make sure that the radical expression of even powers is not negative.
Private values
The root of 0 is 0: .
Root 1 is equal to 1: .
The square root of 0 is 0: .
The square root of 1 is 1: .
Example. Root of roots
Let's look at an example of a square root of roots:
.
Let's transform the inner square root using the formulas above:
.
Now let's transform the original root:
.
So,
.
y = x p for different values of the exponent p.
Here are graphs of the function for non-negative values of the argument x. Graphs of a power function defined for negative values of x are given on the page “Power function, its properties and graphs"
Inverse function
The inverse of a power function with exponent p is a power function with exponent 1/p.
If, then.
Derivative of a power function
Derivative of nth order:
;
Deriving formulas > > >
Integral of a power function
P ≠ - 1
;
.
Power series expansion
At - 1
< x < 1
the following decomposition takes place:
Expressions using complex numbers
Consider the function of the complex variable z:
f (z) = z t.
Let us express the complex variable z in terms of the modulus r and the argument φ (r = |z|):
z = r e i φ .
We represent the complex number t in the form of real and imaginary parts:
t = p + i q .
We have:
Next, we take into account that the argument φ is not uniquely defined:
,
Let's consider the case when q = 0
, that is, the exponent is a real number, t = p. Then
.
If p is an integer, then kp is an integer. Then, due to the periodicity of trigonometric functions:
.
That is, the exponential function with an integer exponent, for a given z, has only one value and is therefore unambiguous.
If p is irrational, then the products kp for any k do not produce an integer. Since k runs through an infinite series of values k = 0, 1, 2, 3, ..., then the function z p has infinitely many values. Whenever the argument z is incremented 2π(one turn), we move to a new branch of the function.
If p is rational, then it can be represented as:
, Where m, n- integers that do not contain common divisors. Then
.
First n values, with k = k 0 = 0, 1, 2, ... n-1, give n different meanings kp:
.
However, subsequent values give values that differ from the previous ones by an integer. For example, when k = k 0+n we have:
.
Trigonometric functions whose arguments differ by multiples of 2π, have equal values. Therefore, with a further increase in k, we obtain the same values of z p as for k = k 0 = 0, 1, 2, ... n-1.
Thus, an exponential function with a rational exponent is multivalued and has n values (branches). Whenever the argument z is incremented 2π(one turn), we move to a new branch of the function. After n such revolutions we return to the first branch from which the countdown began.
In particular, a root of degree n has n values. As an example, consider the nth root of the real positive number z = x. In this case φ 0 = 0 , z = r = |z| = x,
.
.
So, for a square root, n = 2
,
.
For even k, (- 1 ) k = 1. For odd k, (- 1 ) k = - 1.
That is, the square root has two meanings: + and -.
Used literature:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
Lesson and presentation on the topic: "Power functions. Cubic root. Properties of the cubic root"
Additional materials
Dear users, do not forget to leave your comments, reviews, wishes! All materials have been checked by an anti-virus program.
Educational aids and simulators in the Integral online store for grade 9
Educational complex 1C: "Algebraic problems with parameters, grades 9–11" Software environment "1C: Mathematical Constructor 6.0"
Definition of a power function - cube root
Guys, we continue to study power functions. Today we will talk about the "Cubic root of x" function.What is a cube root?
The number y is called a cubic root of x (root of the third degree) if the equality $y^3=x$ holds.
Denoted as $\sqrt(x)$, where x is a radical number, 3 is an exponent.
$\sqrt(27)=3$; $3^3=$27.
$\sqrt((-8))=-2$; $(-2)^3=-8$.
As we can see, the cube root can also be extracted from negative numbers. It turns out that our root exists for all numbers.
The third root of a negative number is negative number. When raised to an odd power, the sign is preserved, the third power is odd.
Let's check the equality: $\sqrt((-x))$=-$\sqrt(x)$.
Let $\sqrt((-x))=a$ and $\sqrt(x)=b$. Let's raise both expressions to the third power. $–x=a^3$ and $x=b^3$. Then $a^3=-b^3$ or $a=-b$. In the notation of roots we obtain the desired identity.
Properties of cube roots
a) $\sqrt(a*b)=\sqrt(a)*\sqrt(6)$.b) $\sqrt(\frac(a)(b))=\frac(\sqrt(a))(\sqrt(b))$.
Let's prove the second property. $(\sqrt(\frac(a)(b)))^3=\frac(\sqrt(a)^3)(\sqrt(b)^3)=\frac(a)(b)$.
We found that the number $\sqrt(\frac(a)(b))$ cubed is equal to $\frac(a)(b)$ and then equals $\sqrt(\frac(a)(b))$, which and needed to be proven.
Guys, let's build a graph of our function.
1) Domain of definition is the set of real numbers.
2) The function is odd, since $\sqrt((-x))$=-$\sqrt(x)$. Next, consider our function for $x≥0$, then display the graph relative to the origin.
3) The function increases when $x≥0$. For our function, a larger value of the argument corresponds to a larger value of the function, which means increase.
4) The function is not limited from above. In fact, we can calculate the third root of an arbitrarily large number, and we can move upward indefinitely, finding ever larger values of the argument.
5) For $x≥0$ the smallest value is 0. This property is obvious.
Let's build a graph of the function by points at x≥0.
Let's construct our graph of the function over the entire domain of definition. Remember that our function is odd. 
Function properties:
1) D(y)=(-∞;+∞).
2) Odd function.
3) Increases by (-∞;+∞).
4) Unlimited.
5) There is no minimum or maximum value.
7) E(y)= (-∞;+∞).
8) Convex downward by (-∞;0), convex upward by (0;+∞).
Examples of solving power functions
Examples1. Solve the equation $\sqrt(x)=x$.
Solution. Let's build two graphs on one coordinate plane$y=\sqrt(x)$ and $y=x$.
As you can see, our graphs intersect at three points. Answer: (-1;-1), (0;0), (1;1).
2. Construct a graph of the function. $y=\sqrt((x-2))-3$.
Solution. Our graph is obtained from the graph of the function $y=\sqrt(x)$, by parallel translation two units to the right and three units down. 
3. Graph the function and read it. $\begin(cases)y=\sqrt(x), x≥-1\\y=-x-2, x≤-1 \end(cases)$.
Solution. Let's construct two graphs of functions on the same coordinate plane, taking into account our conditions. For $x≥-1$ we build a graph of the cubic root, for $x≤-1$ we build a graph of a linear function. 
1) D(y)=(-∞;+∞).
2) The function is neither even nor odd.
3) Decreases by (-∞;-1), increases by (-1;+∞).
4) Unlimited from above, limited from below.
5) Greatest value No. Lowest value equals minus one.
6) The function is continuous on the entire number line.
7) E(y)= (-1;+∞).
Problems to solve independently
1. Solve the equation $\sqrt(x)=2-x$.2. Construct a graph of the function $y=\sqrt((x+1))+1$.
3.Plot a graph of the function and read it. $\begin(cases)y=\sqrt(x), x≥1\\y=(x-1)^2+1, x≤1 \end(cases)$.
Main goals:
1) form an idea of the feasibility of a generalized study of the dependencies of real quantities using the example of quantities related by the relation y=
2) to develop the ability to construct a graph y= and its properties;
3) repeat and consolidate the techniques of oral and written calculations, squaring, extracting square roots.
Equipment, demonstration material: handouts.
1. Algorithm:
2. Sample for completing the task in groups:
3. Sample for self-test of independent work:
4. Card for the reflection stage:
1) I understood how to graph the function y=.
2) I can list its properties using a graph.
3) I did not make mistakes in independent work.
4) I made mistakes in my independent work (list these mistakes and indicate their reason).
Lesson progress
1. Self-determination for educational activities
Purpose of the stage:
1) include students in educational activities;
2) determine the content of the lesson: we continue to work with real numbers.
Organization educational process at stage 1:
– What did we study in the last lesson? (We studied the set of real numbers, operations with them, built an algorithm to describe the properties of a function, repeated functions studied in 7th grade).
– Today we will continue to work with a set of real numbers, a function.
2. Updating knowledge and recording difficulties in activities
Purpose of the stage:
1) update educational content that is necessary and sufficient for the perception of new material: function, independent variable, dependent variable, graphs
y = kx + m, y = kx, y =c, y =x 2, y = - x 2,
2) update mental operations necessary and sufficient for the perception of new material: comparison, analysis, generalization;
3) record all repeated concepts and algorithms in the form of diagrams and symbols;
4) record an individual difficulty in activity, demonstrating at a personally significant level the insufficiency of existing knowledge.
Organization of the educational process at stage 2:
1. Let's remember how you can set dependencies between quantities? (Using text, formula, table, graph)
2. What is a function called? (A relationship between two quantities, where each value of one variable corresponds to a single value of another variable y = f(x)).
What is the name of x? (Independent variable - argument)
What is the name of y? (Dependent variable).
3. In 7th grade did we study functions? (y = kx + m, y = kx, y =c, y =x 2, y = - x 2,).
Individual task:
What is the graph of the functions y = kx + m, y =x 2, y =?
3. Identifying the causes of difficulties and setting goals for activities
Purpose of the stage:
1) organize communicative interaction, during which the distinctive property of the task that caused difficulty in learning activities is identified and recorded;
2) agree on the purpose and topic of the lesson.
Organization of the educational process at stage 3:
-What's special about this task? (The dependence is given by the formula y = which we have not yet encountered.)
– What is the purpose of the lesson? (Get acquainted with the function y =, its properties and graph. Use the function in the table to determine the type of dependence, build a formula and graph.)
– Can you formulate the topic of the lesson? (Function y=, its properties and graph).
– Write the topic in your notebook.
4. Construction of a project for getting out of a difficulty
Purpose of the stage:
1) organize communicative interaction to build a new method of action that eliminates the cause of the identified difficulty;
2) fix new way actions in a symbolic, verbal form and using a standard.
Organization of the educational process at stage 4:
Work at this stage can be organized in groups, asking the groups to build a graph y =, then analyze the results. Groups can also be asked to describe the properties of a given function using an algorithm.
5. Primary consolidation in external speech
The purpose of the stage: to record the studied educational content in external speech.
Organization of the educational process at stage 5:
Construct a graph of y= - and describe its properties.
Properties y= - .
1.Domain of definition of a function.
2. Range of values of the function.
3. y = 0, y> 0, y<0.
y =0 if x = 0.
y<0, если х(0;+)
4.Increasing, decreasing functions.
The function decreases as x.
Let's build a graph of y=.
Let's select its part on the segment. Note that we have = 1 for x = 1, and y max. =3 at x = 9.
Answer: at our name. = 1, y max. =3
6. Independent work with self-test according to the standard
The purpose of the stage: to test your ability to apply new educational content in standard conditions based on comparing your solution with a standard for self-test.
Organization of the educational process at stage 6:
Students complete the task independently, conduct a self-test against the standard, analyze, and correct errors.
Let's build a graph of y=.
Using a graph, find the smallest and largest values of the function on the segment.
7. Inclusion in the knowledge system and repetition
The purpose of the stage: to train the skills of using new content together with previously studied: 2) repeat the educational content that will be required in the following lessons.
Organization of the educational process at stage 7:
Solve the equation graphically: = x – 6.
One student is at the blackboard, the rest are in notebooks.
8. Reflection of activity
Purpose of the stage:
1) record new content learned in the lesson;
2) evaluate your own activities in the lesson;
3) thank classmates who helped get the result of the lesson;
4) record unresolved difficulties as directions for future educational activities;
5) discuss and write down your homework.
Organization of the educational process at stage 8:
- Guys, what was our goal today? (Study the function y=, its properties and graph).
– What knowledge helped us achieve our goal? (Ability to look for patterns, ability to read graphs.)
– Analyze your activities in class. (Cards with reflection)
Homework
paragraph 13 (before example 2) № 13.3, 13.4
Solve the equation graphically:
Construct a graph of the function and describe its properties.
Instead of introducing
The use of modern technologies (CTE) and teaching aids (multimedia board) in lessons helps the teacher plan and conduct effective lessons, create conditions for students to consciously understand, memorize and practice skills.
The lesson turns out to be dynamic and interesting if you combine various forms of teaching during the training session.
In modern didactics, there are four general organizational forms of training:
- individually mediated;
- steam room;
- group;
collective (in shift pairs). (Dyachenko V.K. Modern didactics. - M.: Public Education, 2005).
In a traditional lesson, as a rule, only the first three organizational forms of teaching listed above are used. The collective form of teaching (work in pairs in shifts) is practically not used by the teacher. However, this organizational form of training makes it possible for the team to train everyone and everyone to actively participate in the training of others. The collective form of training is leading in CSR technology.
One of the most common methods of collective learning technology is the “Mutual Training” technique.
This “magic” technique is good in any subject and in any lesson. The purpose is training.
Training is the heir to self-control; it helps the student to establish contact with the subject of study, making it easier to find the right steps and actions. Through training in the acquisition, consolidation, regrouping, revision, and application of knowledge, a person’s cognitive abilities develop. (Yanovitskaya E.V. How to teach and study in a lesson so that you want to learn. Reference album. - St. Petersburg: Educational Projects, M.: Publisher A.M. Kushnir, 2009.-P.14; 131)
It will help you quickly repeat a rule, remember the answers to the questions you have studied, and consolidate the necessary skill. The optimal time to work using the method is 5-10 minutes. As a rule, work on training cards is carried out during oral calculation, that is, at the beginning of the lesson, but at the discretion of the teacher it can be carried out at any stage of the lesson, depending on its goals and structure. A training card can contain from 5 to 10 simple examples (questions, tasks). Each student in the class receives a card. The cards are different for everyone or different for everyone in the “combined squad” (children sitting on the same row). A combined detachment (group) is a temporary cooperation of students formed to perform a specific educational task. (Yalovets T.V. Technology of a collective method of teaching in teacher training: Educational and methodological manual. - Novokuznetsk: IPK Publishing House, 2005. - P. 122)
Lesson project on the topic “Function y=, its properties and graph”
In the lesson project, the topic of which is: “ Function y=, its properties and graph” The use of mutual training techniques in combination with the use of traditional and multimedia teaching tools is presented.
Lesson topic: “ Function y=, its properties and graph”
Goals:
- preparation for the test;
- testing knowledge of all properties of a function and the ability to build graphs of functions and read their properties.
Tasks: subject level:
supra-subject level:
- learn to analyze graphic information;
- practice the ability to conduct dialogue;
- develop the ability to work with an interactive whiteboard using the example of working with graphs.
| Lesson structure | Time |
| 1. Teacher Information Input (TII) | 5 min. |
| 2. Updating basic knowledge: work in shift pairs according to the methodology “Mutual training” | 8 min. |
| 3. Introduction to the topic “Function y=, its properties and graph”: teacher presentation | 8 min. |
| 4. Consolidation of newly learned and already covered material on the topic “Function”: using an interactive whiteboard | 15 min. |
| 5. Self-control : in the form of a test | 7 min. |
| 6. Summing up, recording homework. | 2 min. |
Let us reveal in more detail the content of each stage.
1. Teacher information input (TII) includes an organizational aspect; articulating the topic, purpose and lesson plan; showing a sample of pair work using the mutual training method.
Demonstration of a sample of work in pairs by students at this stage of the lesson is advisable for repeating the algorithm of work of the methodology we need, because At the next stage of the lesson, the work of the entire class team is planned on it. At the same time, you can name the errors in working with the algorithm (if there were any), as well as evaluate the work of these students.
2. Updating of basic knowledge is carried out in shift pairs using the method of mutual training.
The methodology algorithm includes individual, pair (static pairs) and collective (shift pairs) organizational forms of training.
Individual: everyone who receives the card gets acquainted with its contents (reads the questions and answers on the back of the card).
-
first(in the role of the “trainee”) reads the task and answers the questions on the partner’s card;
- second(in the role of “coach”) – checks the correctness of the answers on the back of the card;
- work similarly on another card, changing roles;
- make a mark on an individual sheet and exchange cards;
- move to a new couple.
Collective:
- in the new pair they work like in the first; transition to a new pair, etc.
The number of transitions depends on the time allocated by the teacher for this stage of the lesson, on the diligence and speed of comprehension of each student and on the partners in joint work.
After working in pairs, students make notes on their record sheets, and the teacher conducts a quantitative and qualitative analysis of the work.
The accounting sheet may look like this:
Ivanov Petya 7 “b” grade
| Date | Card number | Number of errors | Who did you work with? |
| 20.12.09 | №7 | 0 | Sidorov K. |
| №3 | 2 | Petrova M. | |
| №2 | 1 | Samoilova Z. |
3. Introduction to the topic “Function y=, its properties and graph” is carried out by the teacher in the form of a presentation using multimedia learning tools (Appendix 4). On the one hand, this is a version of clarity that is understandable to modern students, on the other hand, it saves time on explaining new material.
4. Consolidation of newly learned and already covered material on the topic “Function ” organized in two versions, using traditional teaching tools (blackboard, textbook) and innovative ones (interactive whiteboard).
First, several tasks from the textbook are offered to consolidate the newly learned material. The textbook used for teaching is used. Work is carried out simultaneously with the whole class. In this case, one student completes task “a” - on a traditional board; the other is task “b” on the interactive board, the rest of the students write down the solutions to the same tasks in a notebook and compare their solution with the solution presented on the boards. Next, the teacher evaluates the students’ work at the board.
Then, to more quickly consolidate the studied material on the topic “Function”, frontal work with an interactive whiteboard is proposed, which can be organized as follows:
- the task and schedule appear on the interactive board;
- a student who wants to answer goes to the board, performs the necessary constructions and voices the answer;
- a new task and a new schedule appear on the board;
- Another student comes out to answer.
Thus, in a short period of time, it is possible to solve quite a lot of tasks and evaluate student answers. Some tasks of interest (similar to tasks from the upcoming test) can be recorded in a notebook.
5. At the self-control stage, students are offered a test followed by self-test (Appendix 3).
Literature
- Dyachenko, V.K. Modern didactics [Text] / V.K. Dyachenko - M.: Public education, 2005.
- Yalovets, T.V. Technology of a collective method of teaching in teacher training: Educational and methodological manual [Text] / T.V. Yalovets. – Novokuznetsk: IPK Publishing House, 2005.
- Yanovitskaya, E.V. How to teach and learn in a lesson so that you want to learn. Reference album [Text] / E.V. Yanovitskaya. – St. Petersburg: Educational projects, M.: Publisher A.M. Kushnir, 2009.