Development of the lesson "degree with a natural indicator". Properties of degrees, formulations, proofs, examples Properties of degrees with a natural rule exponent
Technological map of the lesson
Grade 7 Lesson #38
Topic: Degree with a natural indicator
1. Provide repetition, generalization and systematization of knowledge on the topic, consolidate and improve the skills of the simplest transformations of expressions containing degrees with a natural indicator, create conditions for monitoring the assimilation of knowledge and skills;2. To contribute to the formation of skills to apply the methods of generalization, comparison, highlighting the main thing, to promote the education of interest in transferring knowledge to a new situation, the development of mathematical horizons, speech, attention and memory, the development of educational and cognitive activity;
3. To promote the education of interest in mathematics, activity, organization, to cultivate the skills of mutual and self-control of their activities, the formation of a positive motivation for learning, a culture of communication.
Basic concepts of the lesson
Degree, base of a degree, exponent, properties of a degree, product of a degree, division of degrees, raising a degree to a power.
Planned result
They will learn to operate with the concept of Degree, understand the meaning of writing a number as a degree, perform simple transformations of expressions containing degrees with a natural exponent.
They will have the opportunity to learn how to perform transformations of integer expressions containing a degree with a natural exponent
Item Skills, UUD
Personal UUD:
the ability to self-assessment based on the criterion of success of educational activities.
Cognitive UUD:
the ability to navigate in their system of knowledge and skills: to distinguish the new from the already known with the help of a teacher; find answers to questions using the information learned in the lesson.
Generalization and systematization of educational material, operate with a symbolic record of the degree, substitutions, reproduce from memory the information necessary to solve the educational problem
Subject UUD:
Apply degree properties to the transformation of expressions containing powers with a natural exponent
Regulatory UUD:
The ability to determine and formulate the goal in the lesson with the help of a teacher; evaluate your work in class. To exercise mutual control and self-control in the performance of tasks
CommunicativeUUD:
Be able to formulate your thoughts orally and in writing, listen and understand the speech of others
Meta-subject relations
Physics, astronomy, medicine, everyday life
Lesson type
Repetitions, generalizations and application of knowledge and skills.
Forms of work and methods of work
Frontal, steam room, individual. Explanatory - illustrative, verbal, problem situation, workshop, mutual verification, control
Resource support
Components of teaching materials Makarycheva Textbook, projector, screen, computer, presentation, assignments for students, self-assessment sheets
Technologies used in the training session
Technology of semantic reading, problem-based learning, individual and differentiated approach, ICT
Motivate students to work, mobilize attentionGood afternoon guys. Good afternoon, dear colleagues! I greet everyone gathered at today's open lesson. Guys, I want to wish you fruitful work in the lesson, carefully consider the answers to the questions posed, take your time, do not interrupt, respect your classmates and their answers. And I wish you all get only good grades. Good luck to you!
Included in the business rhythm of the lesson
They check the availability of everything necessary for work in the lesson, the accuracy of the location of the Objects. Ability to organize yourself, tune in to work.
2. Actualization of basic knowledge and entry into the topic of the lesson
3. Oral work
Guys, each of you has score sheets on your desk.On them you will evaluate your work in the lesson.Today in the lesson you are given the opportunity to receive not one, but two marks: for work in the lesson and for independent work.
Your correct, complete answers will also be rated "+", but I will put this mark in another column.
On the screen you see puzzles in which the keywords of today's lesson are encrypted. Solve them. (Slide 1)
degree
repetition
generalization
Guys, you correctly guessed the puzzles. These words are degree, repetition, and generalization. And now, using the guessed words - hints, formulate the topic of today's lesson.
Right. Open notebooks and write down the number and topic of the lesson “Repetition and generalization on the topic “Properties of a degree with a natural indicator” (Slide 2)
We have determined the topic of the lesson, but what do you think, what will we do in the lesson, what goals will we set for ourselves? (Slide 3)
Repeat and generalize our knowledge on this topic, fill in the gaps, prepare for the study of the next topic "Monominals".
Guys, the properties of a degree with a natural exponent are quite often used when finding the values of expressions, when converting expressions. The speed of calculations and transformations associated with the properties of a degree with a natural indicator is also dictated by the introduction of the USE.
So, today we will review and summarize your knowledge and skills on this topic. Orally, you must solve a number of problems and remember the verbal grouping of properties and definitions of the degree with a natural indicator.
Epigraph to the lesson of the words of the great Russian scientist M.V. Lomonosov “Let someone try to delete degrees from mathematics, and he will see that without them you won’t go far”
(Slide 4)
Do you think the scientist is right?
Why do we need degrees?
Where are they widely used? (in physics, astronomy, medicine)
That's right, and now let's repeat, what is a degree?
What are the names of a andnin the degree record?
What actions can be performed with degrees? (Slides 5-11)
And now let's sum it up. Do you have worksheets on your desk? .
1. On the left are the beginnings of the definitions, on the right are the endings of the definitions. Connect the correct statements with lines (Slide 12)
Connect the corresponding parts of the definition with lines.a) When multiplying powers with the same base...
1)base degree
b) When dividing powers with the same bases ....
2) Exponent
c) The number a is called
3) the product of n factors, each of which is equal to a.
d) When raising a power to a power...
4) ... the base remains the same, and the indicators add up.
e) The power of a number a with a natural exponent n greater than 1 is called
5) ... the base remains the same, and the indicators are multiplied.
e)Numberncalled
6) Degree
g)Expression a ncalled
7) ... the base remains the same, and the indicators are subtracted.
2.Now, exchange papers with your desk mate, evaluate his work and rate him. Put this score on your score sheet.
Now let's check if you have completed the task correctly.
Guessing puzzles, identifying words - hints.
They are trying to set the topic of the lesson.
Write the date and topic of the lesson in your notebook.
Answer questions
They work in pairs. Read the task, remember.
Connect parts of definitions
They exchange notebooks.
Perform mutual verification of the results, give marks to the neighbor on the desk ..
4. Physical education minute
Hands raised and shook -
these are the trees in the forest,
Hands bent, brushes shaken -
The wind rips off the leaves.
To the sides of the hand, gently wave -
Birds fly south
As they sit down, quietly show -
Hands folded like this!
Perform activities in parallel with the teacher
5. The transfer of acquired knowledge, their primary application in new or changed conditions, in order to form skills.
1. I offer you the following work: you have cards on your desks. You need to complete tasks, i.e. write the answer in the form of a degree with base c, and you will find out the last name and first name of the great French mathematician who introduced the currently generally accepted notation for degrees. (Slide 14)
5WITH 8 : WITH 6
(WITH 4 ) 3 WITH
(WITH 4 ) 3
WITH 4 WITH 5 WITH 0
WITH 5 WITH 3 : WITH 6
WITH 16 : WITH 8
WITH 14 WITH 8
10.
(WITH 3 ) 5
Answer: Rene Descartes.
Story about the biography of Rene Descartes (Slides 15 - 17)
Guys, now let's do the next task.
2. About determine which answers are correct and which are false. (Slide 18 - 19)
Set a true answer to 1, a false answer to 0.
having received an ordered set of ones and zeros, you will find out the correct answer and determine the name and surname of the first Russian woman - mathematician.
a) x 2 x 3 =x 5
b) s 3 s 5 s 8 = s 16
v) x 7 : x 4 = x 28
G) (c+ d) 8 : ( c+ d) 7 = c+ d
e) (x 5 ) 6 = x 30
Choose her name from four names famous women, each of which corresponds to a set of ones and zeros:
Ada Augusta Lovelace - 11001
Sophie Germain - 10101
Ekaterina Dashkova - 11101
Sofia Kovalevskaya - 11011
From the biography of Sofia Kovalevskaya (Slide 20)
Perform the task, determine the surname and name of the French mathematician
Listening and looking at slides
They mark the correct and incorrect answers, write down the resulting code, which determines the name of the first Russian woman - a mathematician.
6. Control and assessment of knowledge Independent performance of tasks by students under the supervision of the teacher.
And now you have to do verification work. You have task cards in front of you. different color. The color corresponds to the level of difficulty of the task (by "3", by "4", by "5") Choose for yourself the task for which grade you will perform and get to work. (Slide 21)
On "3"
1. Express the product as a power:
a) ; b) ;
v) ; G) .
2. Follow these steps:
( m 3 ) 7 ; ( k 4 ) 5 ; (2 2 ) 3; (3 2 ) 5 ; ( m 3 ) 2 ; ( a x ) y
On "4"
1. Present the product as a degree.
a) x 5 X 8 ; boo 2 at 9 ; in 2 6 2 4 ; G)m 2 m 5 m 4 ;
e)x 6 ∙ x 3 ∙ x 7 ; f) (–7) 3 ∙ (–7) 2 ∙ (–7) 9 .
2. Express the quotient as a power:
a)x 8 : x 4 ; b) (–0.5) 10 : (–0,5) 8 ;
c) x 5 : X 3 ; d) 10 : u 10 ; D 2 6 : 2 4 ; e) ;
to "5"
1.Follow the steps:
a) a 4 · a · a 3 a b) (7 X ) 2 c) r · R 2 · R 0
d) with · With 3 · c e) t · T 4 · ( T 2 ) 2 · T 0
e) (2 3 ) 7 : (2 5 ) 3 g) -X 3 · (– X ) 4
h) (R 2 ) 4 : R 5 i)(3 4 ) 2 (3 2 ) 3 : 3 11
2. Simplify:
a) x 3 ( x 2 ) 5 c) ( a 2 ) 3 ( a 4 ) 2
b) ( a 3 ) 2 a 5 g) ( x 2 ) 5 ( x 5 )
Doing assignments in notebooks
7. Lesson summary
Summarizing the information received in the lesson.Checking work, grading. Identification of difficulties encountered in the lesson
8. Reflection
What happened to the concept of degree inXVIIcentury, you and I can predict for ourselves. To do this, try to answer the question: is it possible to raise a number to a negative power or a fractional one? But this is the subject of our future study.
Lesson grades
Guys, I want to finish our lesson with the following parable.
Parable. A wise man was walking, and three people were walking towards him, who were carrying carts with stones for construction under the hot sun. The sage stopped and asked each one a question. He asked the first: “What did you do all day?” And he replied with a grin that he had been carrying cursed stones all day. The sage asked the second: “What did you do all day long”, and he answered: “And I conscientiously did my job.” And the third smiled, his face lit up with joy and pleasure: “And I took part in the construction of the temple!”
Guys, answer, what did you do at the lesson today? Just do it on the self-assessment sheet. Circle the statement in each column that applies to you.
In the self-assessment sheet, you need to underline the phrases that characterize the student's work in the lesson in three areas.
Our lesson is over. Thank you all for your hard work in class!
Answer questions
Evaluate your work in class.
Mark on the card phrases that characterize their work in the lesson.
Lesson topic: Degree with a natural indicatorLesson type: lesson of generalization and systematization of knowledge
Type of lesson: combined
Forms of work: individual, frontal, work in pairs
Equipment: computer, media product (presentation in the programMicrosoftofficepower point 2007); task cards for self-study
Lesson Objectives:
Educational : developing the skills to systematize, generalize knowledge about the degree with a natural indicator, consolidate and improve the skills of the simplest transformations of expressions containing degrees with a natural indicator.
- developing: to promote the formation of skills to apply the methods of generalization, comparison, highlighting the main thing, the development of mathematical horizons, thinking, speech, attention and memory.
- educational: to promote the education of interest in mathematics, activity, organization, to form a positive motive for learning, the development of skills in educational and cognitive activity
Explanatory note.
This lesson is held in a general education class with an average level of mathematical preparation. The main task of the lesson is to develop the skills to systematize, generalize knowledge about the degree with a natural indicator, which is realized in the process of performing various exercises.
The developmental character is manifested in the selection of exercises. The use of a multimedia product allows you to save time, make the material more visual, show samples of design solutions. The lesson uses different kinds works, which relieves the fatigue of children.
Lesson structure:
Organizing time.
Message topics, setting goals for the lesson.
oral work.
Systematization of basic knowledge.
Elements of health-saving technologies.
Execution of a test task
Lesson results.
Homework.
During the classes:
I.Organizing time
Teacher: Hello guys! I am glad to welcome you to our lesson today. Sit down. I hope that today at the lesson we will have both success and joy. And we, working in a team, will show our talent.
Be careful during the lesson. Think, ask, offer - as we will walk the road to truth together.
Open notebooks and write down the number, class work
II. Topic message, lesson goal setting
1) The topic of the lesson. Epigraph of the lesson.(Slide 2.3)
“Let someone try to cross out of mathematics
degree, and he will see that without them you won’t go far” M.V. Lomonosov
2) Setting the objectives of the lesson.
Teacher: So, in the lesson we will repeat, summarize and bring the studied material into the system. Your task is to show your knowledge of the properties of a degree with a natural indicator and the ability to apply them when performing various tasks.
III. Repetition of the basic concepts of the topic, properties of the degree with a natural indicator
1) unravel the anagram: (slide 4)
Nspete (degree)
Whoreosis (cut)
Ovaniosne (base)
Casapotel (indicator)
Multiplication (multiplication)
2) What is a degree with a natural indicator?(Slide 5)
(by the power of the number a with a natural indicator n , greater than 1, is called the expression a n equal to the product n multipliers, each of which is equal to a a-base n -indicator)
3) Read the expression, name the base and the exponent: (Slide 6)
4) Basic properties of the degree (add the right side of the equality)(Slide 7)
a n a m =
a n :a m =
(a n ) m =
(ab) n =
( a / b ) n =
a 0 =
a 1 =
IV At stnaya Work
1) verbal account (slide8)
Teacher: And now let's check how you can apply these formulas when solving.
1)x 5 X 7 ; 2) a 4 a 0 ;
3) to 9 : To 7 ; 4) r n : r ;
5)5 5 2 ; 6) (- b )(- b ) 3 (- b );
7) with 4 : With; 8) 7 3 : 49;
9) 4 at 6 y 10) 7 4 49 7 3 ;
11) 16: 4 2 ; 12) 64: 8 2 ;
13) sss 3 ; 14) a 2 n a n ;
15) x 9 : X m ; 16) at n : u
2) the game "Exclude the excess" ((-1) 2 )(slide9)
-1
Well done. They did a good job. We then solve the following examples.
VSystematization of basic knowledge
1. Connect the expressions corresponding to each other with lines:(slide 10)
4 ⁶ 4 2 3 6 4 6
4 6 : 4 2 4 6 /5 6
(3 4) 6 4 ⁶ +2
(4 2 ) 6 4 6-2
(4/5) 6 4 12
2. Arrange in ascending order of the number:(slide 11)
3 2 (-0.5) 3 (½) 3 35 0 (-10) 3
3. Completion of the task with subsequent self-examination(slide 12)
A1 represent the product in the form of a degree:
a) a) x 5 X 4 ; b) 3 7 3 9 ; at 4) 3 (-4) 8 .
And 2 simplify the expression:
a) x 3 X 7 X 8 ; b) 2 21 :2 19 2 3
And 3 exponentiate:
a) (a 5 ) 3 ; b) (-c 7 ) 2
VIElements of health-saving technologies (slide 13)
Physical education: repetition of the degree of numbers 2 and 3
VIITest task (slide 14)
Answers to the test are written on the board: 1 d 2 o 3b 4s 5 h 6a (extraction)
VIII Independent work on cards
On each desk, cards with a task according to options, after the work is completed, they are submitted for verification
Option 1
1) Simplify expressions:
a) 
b) 
v) 
G) 
a) 
b) 
v) 
G)
Option 2
1) Simplify expressions:
a) b)
v) 
G)
2) Find the value of the expression:
a)b)
v) 
G) 
3) Show with an arrow whether the value of the expression is equal to zero, a positive or negative number:
IX Lesson summary
No. p / p
Type of work
self-esteem
Teacher evaluation
1
Anagram
2
Read the expression
3
Rules
4
Verbal counting
5
Connect with lines
6
Arrange in ascending order
7
Self-test tasks
8
Test
9
Independent work on cards
X Homework
Test cards
A1. Find the value of the expression: .
Earlier we already talked about what a power of a number is. It has certain properties that are useful in solving problems: it is them and all possible exponents that we will analyze in this article. We will also demonstrate with examples how they can be proved and correctly applied in practice.
Let us recall the concept of a degree with a natural exponent, which we have already formulated earlier: this is the product of the nth number of factors, each of which is equal to a. We also need to remember how to correctly multiply real numbers. All this will help us to formulate the following properties for a degree with a natural indicator:
Definition 1
1. The main property of the degree: a m a n = a m + n
Can be generalized to: a n 1 · a n 2 · … · a n k = a n 1 + n 2 + … + n k .
2. The quotient property for powers that have the same base: a m: a n = a m − n
3. Product degree property: (a b) n = a n b n
The equality can be extended to: (a 1 a 2 … a k) n = a 1 n a 2 n … a k n
4. Property of a natural degree: (a: b) n = a n: b n
5. We raise the power to the power: (a m) n = a m n ,
Can be generalized to: (((a n 1) n 2) …) n k = a n 1 n 2 … n k
6. Compare the degree with zero:
- if a > 0, then for any natural n, a n will be greater than zero;
- with a equal to 0, a n will also be equal to zero;
- for a< 0 и таком показателе степени, который будет четным числом 2 · m , a 2 · m будет больше нуля;
- for a< 0 и таком показателе степени, который будет нечетным числом 2 · m − 1 , a 2 · m − 1 будет меньше нуля.
7. Equality a n< b n будет справедливо для любого натурального n при условии, что a и b больше нуля и не равны друг другу.
8. The inequality a m > a n will be true provided that m and n are integers, m is greater than n and a is greater than zero and not less than one.
As a result, we got several equalities; if you meet all the conditions indicated above, then they will be identical. For each of the equalities, for example, for the main property, you can swap the right and left parts: a m · a n = a m + n - the same as a m + n = a m · a n . In this form, it is often used when simplifying expressions.
1. Let's start with the main property of the degree: the equality a m · a n = a m + n will be true for any natural m and n and real a . How to prove this statement?
The basic definition of powers with natural exponents will allow us to convert equality into a product of factors. We will get an entry like this:
This can be shortened to 
(recall the basic properties of multiplication). As a result, we got the degree of the number a with natural exponent m + n. Thus, a m + n , which means that the main property of the degree is proved.
Let's take a concrete example to prove this.
Example 1
So we have two powers with base 2. Their natural indicators are 2 and 3, respectively. We got the equality: 2 2 2 3 = 2 2 + 3 = 2 5 Let's calculate the values to check the correctness of this equality.
Let's perform the necessary mathematical operations: 2 2 2 3 = (2 2) (2 2 2) = 4 8 = 32 and 2 5 = 2 2 2 2 2 = 32
As a result, we got: 2 2 2 3 = 2 5 . The property has been proven.
Due to the properties of multiplication, we can generalize the property by formulating it in the form of three or more powers, for which the exponents are natural numbers, and the bases are the same. If we denote the number of natural numbers n 1, n 2, etc. by the letter k, we get the correct equality:
a n 1 a n 2 … a n k = a n 1 + n 2 + … + n k .
Example 2
2. Next, we need to prove the following property, which is called the quotient property and is inherent in powers with the same base: this is the equality am: an = am − n , which is valid for any natural m and n (and m is greater than n)) and any non-zero real a .
To begin with, let us explain what exactly is the meaning of the conditions that are mentioned in the formulation. If we take a equal to zero, then in the end we will get a division by zero, which cannot be done (after all, 0 n = 0). The condition that the number m must be greater than n is necessary so that we can stay within the natural exponents: by subtracting n from m, we get a natural number. If the condition is not met, we will get a negative number or zero, and again we will go beyond the study of degrees with natural indicators.
Now we can move on to the proof. From the previously studied, we recall the basic properties of fractions and formulate the equality as follows:
a m − n a n = a (m − n) + n = a m
From it we can deduce: a m − n a n = a m
Recall the connection between division and multiplication. It follows from it that a m − n is a quotient of powers a m and a n . This is the proof of the second degree property.
Example 3
Substitute specific numbers for clarity in indicators, and denote the base of the degree π: π 5: π 2 = π 5 − 3 = π 3
3. Next, we will analyze the property of the degree of the product: (a · b) n = a n · b n for any real a and b and natural n .
According to the basic definition of a degree with a natural exponent, we can reformulate the equality as follows:
Remembering the properties of multiplication, we write: 
. It means the same as a n · b n .
Example 4
2 3 - 4 2 5 4 = 2 3 4 - 4 2 5 4
If we have three or more factors, then this property also applies to this case. We introduce the notation k for the number of factors and write:
(a 1 a 2 … a k) n = a 1 n a 2 n … a k n
Example 5
With specific numbers, we get the following correct equality: (2 (- 2 , 3) a) 7 = 2 7 (- 2 , 3) 7 a
4. After that, we will try to prove the quotient property: (a: b) n = a n: b n for any real a and b if b is not equal to 0 and n is a natural number.
For the proof, we can use the previous degree property. If (a: b) n bn = ((a: b) b) n = an , and (a: b) n bn = an , then it follows that (a: b) n is a quotient of dividing an by bn .
Example 6
Let's count the example: 3 1 2: - 0 . 5 3 = 3 1 2 3: (- 0 , 5) 3
Example 7
Let's start right away with an example: (5 2) 3 = 5 2 3 = 5 6
And now we formulate a chain of equalities that will prove to us the correctness of the equality: 
If we have degrees of degrees in the example, then this property is true for them as well. If we have any natural numbers p, q, r, s, then it will be true:
a p q y s = a p q y s
Example 8
Let's add specifics: (((5 , 2) 3) 2) 5 = (5 , 2) 3 2 5 = (5 , 2) 30
6. Another property of degrees with a natural exponent that we need to prove is the comparison property.
First, let's compare the exponent with zero. Why a n > 0 provided that a is greater than 0?
If we multiply one positive number by another, we will also get a positive number. Knowing this fact, we can say that this does not depend on the number of factors - the result of multiplying any number of positive numbers is a positive number. And what is a degree, if not the result of multiplying numbers? Then for any power a n with a positive base and a natural exponent, this will be true.
Example 9
3 5 > 0 , (0 , 00201) 2 > 0 and 34 9 13 51 > 0
It is also obvious that a power with a base equal to zero is itself zero. To whatever power we raise zero, it will remain so.
Example 10
0 3 = 0 and 0 762 = 0
If the base of the degree is a negative number, then the proof is a little more complicated, since the concept of even / odd exponent becomes important. Let's start with the case when the exponent is even and denote it by 2 · m , where m is a natural number.
Let's remember how to correctly multiply negative numbers: the product a · a is equal to the product of modules, and, therefore, it will be a positive number. Then 
and the degree a 2 · m are also positive.
Example 11
For example, (− 6) 4 > 0 , (− 2 , 2) 12 > 0 and - 2 9 6 > 0
What if the exponent with a negative base is an odd number? Let's denote it 2 · m − 1 .
Then 
All products a · a , according to the properties of multiplication, are positive, and so is their product. But if we multiply it by the only remaining number a , then the final result will be negative.
Then we get: (− 5) 3< 0 , (− 0 , 003) 17 < 0 и - 1 1 102 9 < 0
How to prove it?
a n< b n – неравенство, представляющее собой произведение левых и правых частей nверных неравенств a < b . Вспомним основные свойства неравенств справедливо и a n < b n .
Example 12
For example, the inequalities are true: 3 7< (2 , 2) 7 и 3 5 11 124 > (0 , 75) 124
8. It remains for us to prove the last property: if we have two degrees, the bases of which are the same and positive, and the exponents are natural numbers, then the one of them is greater, the exponent of which is less; and of two degrees with natural indicators and the same bases greater than one, the degree whose indicator is greater is greater.
Let's prove these assertions.
First we need to make sure that a m< a n при условии, что m больше, чем n , и а больше 0 , но меньше 1 .Теперь сравним с нулем разность a m − a n
We take a n out of brackets, after which our difference will take the form a n · (am − n − 1) . Its result will be negative (since the result of multiplying a positive number by a negative one is negative). Indeed, according to the initial conditions, m − n > 0, then a m − n − 1 is negative, and the first factor is positive, like any natural power with a positive base.
It turned out that a m − a n< 0 и a m < a n . Свойство доказано.
It remains to prove the second part of the statement formulated above: a m > a is true for m > n and a > 1 . We indicate the difference and take a n out of brackets: (a m - n - 1). The power of a n for a greater than one will give positive result; and the difference itself will also turn out to be positive due to the initial conditions, and for a > 1 the degree of a m − n is greater than one. It turns out that a m − a n > 0 and a m > a n , which is what we needed to prove.
Example 13
Example with specific numbers: 3 7 > 3 2
Basic properties of degrees with integer exponents
For degrees with positive integer exponents, the properties will be similar, because positive integers are natural numbers, which means that all the equalities proved above are also valid for them. They are also suitable for cases where the exponents are negative or equal to zero (provided that the base of the degree itself is non-zero).
Thus, the properties of powers are the same for any bases a and b (provided that these numbers are real and not equal to 0) and any exponents m and n (provided that they are integers). We write them briefly in the form of formulas:
Definition 2
1. a m a n = a m + n
2. a m: a n = a m − n
3. (a b) n = a n b n
4. (a: b) n = a n: b n
5. (am) n = a m n
6. a n< b n и a − n >b − n with positive integer n , positive a and b , a< b
7. a m< a n , при условии целых m и n , m >n and 0< a < 1 , при a >1 a m > a n .
If the base of the degree is equal to zero, then the entries a m and a n make sense only in the case of natural and positive m and n. As a result, we find that the formulations above are also suitable for cases with a degree with a zero base, if all other conditions are met.
The proofs of these properties in this case are simple. We will need to remember what a degree with a natural and integer exponent is, as well as the properties of actions with real numbers.
Let us analyze the property of the degree in the degree and prove that it is true for both positive integers and non-positive integers. We start by proving the equalities (ap) q = ap q , (a − p) q = a (− p) q , (ap) − q = ap (− q) and (a − p) − q = a (−p) (−q)
Conditions: p = 0 or natural number; q - similarly.
If the values of p and q are greater than 0, then we get (a p) q = a p · q . We have already proved a similar equality before. If p = 0 then:
(a 0) q = 1 q = 1 a 0 q = a 0 = 1
Therefore, (a 0) q = a 0 q
For q = 0 everything is exactly the same:
(a p) 0 = 1 a p 0 = a 0 = 1
Result: (a p) 0 = a p 0 .
If both indicators are zero, then (a 0) 0 = 1 0 = 1 and a 0 0 = a 0 = 1, then (a 0) 0 = a 0 0 .
Recall the property of the quotient in the power proved above and write:
1 a p q = 1 q a p q
If 1 p = 1 1 … 1 = 1 and a p q = a p q , then 1 q a p q = 1 a p q
We can transform this notation by virtue of the basic multiplication rules into a (− p) · q .
Also: a p - q = 1 (a p) q = 1 a p q = a - (p q) = a p (- q) .
AND (a - p) - q = 1 a p - q = (a p) q = a p q = a (- p) (- q)
The remaining properties of the degree can be proved in a similar way by transforming the existing inequalities. We will not dwell on this in detail, we will only indicate the difficult points.
Proof of the penultimate property: recall that a − n > b − n is true for any negative integer values of n and any positive a and b, provided that a is less than b .
Then the inequality can be transformed as follows:
1 a n > 1 b n
We write the right and left parts as a difference and perform the necessary transformations:
1 a n - 1 b n = b n - a n a n b n
Recall that in the condition a is less than b , then, according to the definition of a degree with a natural exponent: - a n< b n , в итоге: b n − a n > 0 .
a n · b n ends up being a positive number because its factors are positive. As a result, we have a fraction b n - a n a n · b n , which in the end also gives a positive result. Hence 1 a n > 1 b n whence a − n > b − n , which we had to prove.
The last property of degrees with integer exponents is proved similarly to the property of degrees with natural exponents.
Basic properties of degrees with rational exponents
In previous articles, we discussed what a degree with a rational (fractional) exponent is. Their properties are the same as those of degrees with integer exponents. Let's write:
Definition 3
1. am 1 n 1 am 2 n 2 = am 1 n 1 + m 2 n 2 for a > 0, and if m 1 n 1 > 0 and m 2 n 2 > 0, then for a ≥ 0 (product property powers with the same base).
2. a m 1 n 1: b m 2 n 2 = a m 1 n 1 - m 2 n 2 if a > 0 (quotient property).
3. a bmn = amn bmn for a > 0 and b > 0, and if m 1 n 1 > 0 and m 2 n 2 > 0, then for a ≥ 0 and (or) b ≥ 0 (product property in fractional degree).
4. a: b m n \u003d a m n: b m n for a > 0 and b > 0, and if m n > 0, then for a ≥ 0 and b > 0 (property of a quotient to a fractional degree).
5. am 1 n 1 m 2 n 2 = am 1 n 1 m 2 n 2 for a > 0, and if m 1 n 1 > 0 and m 2 n 2 > 0, then for a ≥ 0 (degree property in degrees).
6.ap< b p при условии любых положительных a и b , a < b и рациональном p при p >0; if p< 0 - a p >b p (the property of comparing degrees with equal rational exponents).
7.ap< a q при условии рациональных чисел p и q , p >q at 0< a < 1 ; если a >0 – a p > a q
To prove these statements, we need to recall what a degree with fractional indicator what are the properties arithmetic root n-th degree and what are the properties of a degree with integer exponents. Let's take a look at each property.
According to what a degree with a fractional exponent is, we get:
a m 1 n 1 \u003d am 1 n 1 and a m 2 n 2 \u003d am 2 n 2, therefore, a m 1 n 1 a m 2 n 2 \u003d am 1 n 1 a m 2 n 2
The properties of the root will allow us to derive equalities:
a m 1 m 2 n 1 n 2 a m 2 m 1 n 2 n 1 = a m 1 n 2 a m 2 n 1 n 1 n 2
From this we get: a m 1 n 2 a m 2 n 1 n 1 n 2 = a m 1 n 2 + m 2 n 1 n 1 n 2
Let's transform:
a m 1 n 2 a m 2 n 1 n 1 n 2 = a m 1 n 2 + m 2 n 1 n 1 n 2
The exponent can be written as:
m 1 n 2 + m 2 n 1 n 1 n 2 = m 1 n 2 n 1 n 2 + m 2 n 1 n 1 n 2 = m 1 n 1 + m 2 n 2
This is the proof. The second property is proved in exactly the same way. Let's write down the chain of equalities:
am 1 n 1: am 2 n 2 = am 1 n 1: am 2 n 2 = am 1 n 2: am 2 n 1 n 1 n 2 = = am 1 n 2 - m 2 n 1 n 1 n 2 \u003d am 1 n 2 - m 2 n 1 n 1 n 2 \u003d am 1 n 2 n 1 n 2 - m 2 n 1 n 1 n 2 \u003d am 1 n 1 - m 2 n 2
Proofs of the remaining equalities:
a b m n = (a b) m n = a m b m n = a m n b m n = a m n b m n ; (a: b) m n = (a: b) m n = a m: b m n = = a m n: b m n = a m n: b m n ; am 1 n 1 m 2 n 2 = am 1 n 1 m 2 n 2 = am 1 n 1 m 2 n 2 = = am 1 m 2 n 1 n 2 = am 1 m 2 n 1 n 2 = = am 1 m 2 n 2 n 1 = am 1 m 2 n 2 n 1 = am 1 n 1 m 2 n 2
Next property: let's prove that for any values of a and b greater than 0 , if a is less than b , a p will be executed< b p , а для p больше 0 - a p >bp
Let's represent a rational number p as m n . In this case, m is an integer, n is a natural number. Then the conditions p< 0 и p >0 will be extended to m< 0 и m >0 . For m > 0 and a< b имеем (согласно свойству степени с целым положительным показателем), что должно выполняться неравенство a m < b m .
We use the property of roots and derive: a m n< b m n
Taking into account the positiveness of the values a and b , we rewrite the inequality as a m n< b m n . Оно эквивалентно a p < b p .
In the same way, for m< 0 имеем a a m >b m , we get a m n > b m n so a m n > b m n and a p > b p .
It remains for us to prove the last property. Let us prove that for rational numbers p and q , p > q at 0< a < 1 a p < a q , а при a >0 would be true a p > a q .
The rational numbers p and q can be reduced to common denominator and get fractions m 1 n and m 2 n
Here m 1 and m 2 are integers, and n is a natural number. If p > q, then m 1 > m 2 (taking into account the rule for comparing fractions). Then at 0< a < 1 будет верно a m 1 < a m 2 , а при a >1 – inequality a 1 m > a 2 m .
They can be rewritten in the following form:
a m 1 n< a m 2 n a m 1 n >a m 2 n
Then you can make transformations and get as a result:
a m 1 n< a m 2 n a m 1 n >a m 2 n
To summarize: for p > q and 0< a < 1 верно a p < a q , а при a >0 – a p > a q .
Basic properties of degrees with irrational exponents
All the properties described above that a degree with rational exponents possesses can be extended to such a degree. This follows from its very definition, which we gave in one of the previous articles. Let us briefly formulate these properties (conditions: a > 0 , b > 0 , exponents p and q are irrational numbers):
Definition 4
1. a p a q = a p + q
2. a p: a q = a p − q
3. (a b) p = a p b p
4. (a: b) p = a p: b p
5. (a p) q = a p q
6.ap< b p верно при любых положительных a и b , если a < b и p – иррациональное число больше 0 ; если p меньше 0 , то a p >bp
7.ap< a q верно, если p и q – иррациональные числа, p < q , 0 < a < 1 ; если a >0 , then a p > a q .
Thus, all powers whose exponents p and q are real numbers, provided that a > 0, have the same properties.
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
algebra 7th grade
mathematic teacher
branch MBOUTSOSH №1
in the village of Poletaevo Zueva I.P.
Poletaevo 2016
Topic: « Properties of degree with natural exponent»
PURPOSE
- Repetition, generalization and systematization of the studied material on the topic "Properties of a degree with a natural indicator".
- Checking students' knowledge on the topic.
- Application of the acquired knowledge in the performance of various tasks.
TASKS
subject :
repeat, generalize and systematize knowledge on the topic; create conditions for control (mutual control) of the assimilation of knowledge and skills;continue the formation of students' motivation to study the subject;
metasubject:
develop an operational style of thinking; to promote the acquisition of communication skills by students when working together; activate their creative thinking; Pto continue the formation of certain competencies of students, which will contribute to their effective socialization;skills of self-education and self-education.
personal:
educate culture, promote the formation of personal qualities aimed at a benevolent, tolerant attitude towards each other, people, life; to cultivate initiative and independence in activities; lead to an understanding of the need for the topic under study for successful preparation for the state final certification.
LESSON TYPE
generalization and systematization lesson ZUN.
Equipment: computer, projector,projection screen,board, handout.
Software: Windows 7 OS: MS Office 2007 (required application - powerpoint).
Preparatory stage:
presentation "Properties of a degree with a natural indicator";
Handout;
score sheet.
Structure
Organizing time. Setting goals and objectives of the lesson - 3 minutes.
Actualization, systematization of basic knowledge - 8 minutes.
Practical part - 28 minutes.
Generalization, conclusion -3 minutes.
Homework- 1 minute.
Reflection - 2 minutes.
Lesson idea
Checking in an interesting and effective way the ZUN of students on this topic.
Organization of the lesson The lesson is held in the 7th grade. The children work in pairs, independently, the teacher acts as a consultant-observer.
During the classes
Organizing time:
Hello guys! Today we have an unusual lesson-game. Each of you is given a great opportunity to prove yourself, show your knowledge. Perhaps during the lesson you will discover hidden abilities in yourself that will be useful to you in the future.
Each of you has a test sheet and cards for completing tasks in them. Take a test sheet in your hands, you need it so that you yourself evaluate your knowledge during the lesson. Sign it.
So, I invite you to the lesson!
Guys, look at the screen and listen to the poem.
Slide #1
Multiply and divide
Raising a power to a power...
We are familiar with these properties.
And they are no longer new.
These five simple rules
Everyone in the class already answered
But if you forgot the properties,
Consider the example you did not solve!
And in order to live without troubles at school
I'll give you good advice:
Do you want to forget the rule?
Just try to learn!
Answer the question:
1) What actions are mentioned in it?
2) What do you think we will talk about today at the lesson?
So the topic of our lesson is:
"Properties of a degree with a natural exponent" (Slide 3).
Setting goals and objectives of the lesson
In the lesson, we will repeat, summarize and bring into the system the studied material on the topic “Properties of a degree with a natural indicator”
Let's see how you learned how to multiply and divide powers with the same base, as well as raise a power to a power
Updating of basic knowledge. Systematization of theoretical material.
1) Oral work
Let's work verbally
1) Formulate the properties of the degree with a natural indicator.
2) Fill in the blanks: (Slide 4)
1)5 12 : 5 5 =5 7 2) 5 7 ∙ 5 17 = 5 24 3) 5 24 : 125= 5 21 4)(5 0 ) 2 ∙5 24 =5 24
5)5 12 ∙ 5 12 = (5 8 ) 3 6)(3 12 ) 2 = 3 24 7) 13 0 ∙ 13 64 = 13 64
3) What is the value of the expression:(Slide 5-9)
a m ∙ a n; (a m+n ) a m : a n (a m-n ) ; (a m ) n ; a 1; and 0 .
2) Checking the theoretical part (Card #1)
Now pick up card number 1 andfill the gaps
1) If the indicator is an even number, then the value of the degree is always _______________
2) If the indicator is an odd number, then the value of the degree coincides with the sign ____.
3) Product of powers a n a k = a n + k
When multiplying powers with the same base, you need the base ____________, and the exponents ________.
4)Private degrees a n : a k = a n - k
When dividing powers with the same base, you need the base _____, and from the indicator of the dividend ____________________________.
5) Raising a degree to a power ( a n ) k = a nk
When raising a power to a power, the base is _______, and the exponents are ______.
Checking answers. (Slides 10-13)
Main part
3) And now we open notebooks, write down the number 28.01 14g, class work
The game "Clapperboard" » (Slide 14)
Complete the assignments in your notebooks on your own
Do the following: a)X11 ∙х∙х2 b)X14 : X5 c) (a4 ) 3 d) (-For)2 .
Compare the value of the expression with zero: a) (- 5)7 , b)(-6)18 ,
at 4)11 . ( -4) 8 G)(- 5) 18 ∙ (- 5) 6 , e)-(- 4)8 .
Calculate the value of an expression:
a) -1 ∙ 3 2 , b) (-1 ∙ 3) 2 c) 1 ∙ (-3) 2 , d) - (2 ∙ 3) 2 , e) 1 2 ∙ (-3) 2
We check, if the answer is not correct, we make one clap of our hands.
Calculate the number of points and put them on the score sheet.
4) And now we will do gymnastics for the eyes, relieve tension, and we will continue to work. We carefully monitor the movement of objects
Begin! (Slide 15,16,17,18).
5) And now let's proceed to the next type of our work. (Card2)
Write your answer as a power with a base WITH and you will learn the name and surname of the great French mathematician who was the first to introduce the concept of the degree of a number.
Guess the name of the scientist mathematician.
| 1. | WITH 5 ∙С 3 | 6. | WITH 7 : WITH 5 |
| 2. | WITH 8 : WITH 6 | 7. | (WITH 4 ) 3 ∙С |
| 3, | (WITH 4 ) 3 | 8. | WITH 4 ∙ WITH 5 ∙ C 0 |
| 4. | WITH 5 ∙С 3 : WITH 6 | 9. | WITH 16 : WITH 8 |
| 5. | WITH 14 ∙ C 8 | 10. | (WITH 3 ) 5 |
O Answer: RENE DECARTES
| R | W | M | YU | TO | H | A | T | E | D |
|||||
| WITH 8 | WITH 5 | WITH 1 | WITH 40 | WITH 13 | WITH 12 | WITH 9 | WITH 15 | WITH 2 | WITH 22 |
And now let's listen to the student's message about "Rene Descartes"
Rene Descartes was born on March 21, 1596 in the small town of La Gaie in Touraine. The Descartes family belonged to the humble bureaucratic nobility. Rene spent his childhood in Touraine. Descartes finished school in 1612. He spent eight and a half years there. Descartes did not immediately find his place in life. A nobleman by birth, after graduating from college in La Fleche, he plunges headlong into social life Paris, then leaves everything for the sake of science. Descartes gave mathematics a special place in his system, he considered its principles of establishing the truth a model for other sciences. A considerable merit of Descartes was the introduction of convenient designations that have survived to this day: the Latin letters x, y, z for unknowns; a, c, c - for coefficients, for degrees. Descartes' interests are not limited to mathematics, but include mechanics, optics, and biology. In 1649 Descartes, after long hesitation, moved to Sweden. This decision turned out to be fatal for his health. Six months later, Descartes died of pneumonia.
6) Work at the board:
1. Solve the Equation
A) x 4 ∙ (x 5) 2 / x 20: x 8 \u003d 49
B) (t 7 ∙ t 17 ) : (t 0 ∙ t 21 )= -125
2.Calculate the value of the expression:
(5-x) 2 -2x 3 +3x 2 -4x+x-x 0
a) at x=-1
b) at x=2 Independently
7) Take card number 3 in your hands, do the test
| Option 1 | Option 2. |
| 1. Do the division of powers 2 17 : 2 5 2 12 2 45 2. Write in the form of a degree (x + y) (x + y) \u003d x 2 + y 2 (x+y) 2 2(x+y) 3. Replace * degree so that the equality a 5 · * =a 15 a 10 a 3 (a 7 ) 5 ? a) a 12 b) a 5 c) a 35 3 = 8 15 8 12 6. Find the value of the fraction | 1. Do division by powers of 9 9 : 9 7 9 16 9 63 2. Write in the form of a degree (x-y) (x-y) \u003d ... x 2 -y 2 (x-y) 2 2(x-y) 3. Replace * degree so that the equality b 9 · * = b 18 b 17 b 1 1 4. What is the value of the expression(with 6 ) 4 ? a) from 10 b) from 6 c) since 24 5. From the proposed options, choose the one that can replace * in equality (*) 3 = 5 24 5 21 6. Find the value of the fraction |
Check each other's work and rate your comrades on the grade sheet.
| 1 option | a | b | b | With | b | 3 |
| Option 2 | a | b | With | With | a | 4 |
Additional tasks for strong learners
Each task is evaluated separately.
Find the value of an expression:
8) And now let's see the effectiveness of our lesson ( Slide 19)
To do this, complete the task, cross out the letters corresponding to the answers.
AOWSTLCCRCHGNMO
Simplify the expression:
| 1. | С 4 ∙С 3 | 5. | (WITH 2 ) 3 ∙ WITH 5 |
| 2. | (C 5 ) 3 | 6. | WITH 6 ∙ WITH 5 : WITH 10 |
| 3. | From 11 : From 6 | 7. | (WITH 4 ) 3 ∙С 2 |
| 4. | C 5 ∙C 5 : C |
Cipher: A - From 7 V- From 15 G - WITH AND - From 30 TO - From 9 M - From 14 H - From 13 O - From 12 R - From 11 WITH - From 5 T - From 8 H - From 3
What word did you get? ANSWER: EXCELLENT! (Slide 20)
Summing up, evaluation, marking (Slide 21)
Let's summarize our lesson, how successfully we repeated, generalized and systematized knowledge on the topic "Properties of a degree with a natural indicator"
We take test sheets and calculate the total number of points and write them down in the line of the final grade
Stand up who scored 29-32 points: excellent score
25-28 points: score - good
20-24 points: score - satisfactory
I will once again check the correctness of the assignments on the cards, check your results with the points set in the test sheet. I will put the grades in the journal
And for active work in the assessment lesson:
Children, I ask you to evaluate your work in the lesson. Mark on the mood sheet.
| Test sheet |
||
| Last name First name | Grade |
|
| 1. Theoretical part | ||
| 2. Game "Clapperboard" | ||
| 3. Test | ||
| 4. "Cipher" | ||
| Additional part | ||
| Final grade: | ||
| Emotional evaluation | About myself | About the lesson |
| Satisfied | ||
| dissatisfied | ||
Homework (Slide 22)
Make a crossword puzzle with the keyword DEGREE. In the next lesson, we will look at the most interesting works.
№ 567
List of sources used
- Textbook "Algebra Grade 7".
- Poem. http://yandex.ru/yandsearch
- NOT. Shchurkov. The culture of the modern lesson. Moscow: Russian Pedagogical Agency, 1997.
- A.V. Petrov. Methodological and methodological foundations personality-developing computer education. Volgograd. "Change", 2001.
- A.S. Belkin. situation of success. How to create it. M .: "Enlightenment", 1991.
- Computer science and education №3. Operational Thinking Style, 2003
After the degree of the number is determined, it is logical to talk about degree properties. In this article, we will give the basic properties of the degree of a number, while touching on all possible exponents. Here we will give proofs of all properties of the degree, and also show how these properties are applied when solving examples.
Page navigation.
Properties of degrees with natural indicators
By definition of a power with a natural exponent, the power of a n is the product of n factors, each of which is equal to a . Based on this definition, and using real number multiplication properties, we can obtain and justify the following properties of degree with natural exponent:
- the main property of the degree a m ·a n =a m+n , its generalization ;
- the property of partial powers with the same bases a m:a n =a m−n ;
- product degree property (a b) n =a n b n , its extension ;
- quotient property in kind (a:b) n =a n:b n ;
- exponentiation (a m) n =a m n , its generalization (((a n 1) n 2) ...) n k \u003d a n 1 n 2 ... n k;
- comparing degree with zero:
- if a>0 , then a n >0 for any natural n ;
- if a=0 , then a n =0 ;
- if a<0 и показатель степени является четным числом 2·m , то a 2·m >0 if a<0 и показатель степени есть нечетное число 2·m−1 , то a 2·m−1 <0 ;
- if a and b are positive numbers and a
- if m and n are natural numbers such that m>n , then at 0
- if m and n are natural numbers such that m>n , then at 0
We immediately note that all the written equalities are identical under the specified conditions, and their right and left parts can be interchanged. For example, the main property of the fraction a m a n = a m + n with simplification of expressions often used in the form a m+n = a m a n .
Now let's look at each of them in detail.
Let's start with the property of the product of two powers with the same bases, which is called the main property of the degree: for any real number a and any natural numbers m and n, the equality a m ·a n =a m+n is true.
Let us prove the main property of the degree. By the definition of a degree with a natural exponent, the product of powers with the same bases of the form a m a n can be written as a product. Due to the properties of multiplication, the resulting expression can be written as 
, and this product is the power of a with natural exponent m+n , that is, a m+n . This completes the proof.
Let us give an example that confirms the main property of the degree. Let's take degrees with the same bases 2 and natural powers 2 and 3, according to the main property of the degree, we can write the equality 2 2 ·2 3 =2 2+3 =2 5 . Let's check its validity, for which we calculate the values of the expressions 2 2 ·2 3 and 2 5 . Performing exponentiation, we have 2 2 2 3 =(2 2) (2 2 2)=4 8=32 and 2 5 \u003d 2 2 2 2 2 \u003d 32, since equal values are obtained, then the equality 2 2 2 3 \u003d 2 5 is correct, and it confirms the main property of the degree.
The main property of a degree based on the properties of multiplication can be generalized to the product of three or more degrees with the same bases and natural exponents. So for any number k of natural numbers n 1 , n 2 , …, n k the equality a n 1 a n 2 a n k =a n 1 +n 2 +…+n k.
For instance, (2.1) 3 (2.1) 3 (2.1) 4 (2.1) 7 = (2,1) 3+3+4+7 =(2,1) 17 .
You can move on to the next property of degrees with a natural indicator - the property of partial powers with the same bases: for any non-zero real number a and arbitrary natural numbers m and n satisfying the condition m>n , the equality a m:a n =a m−n is true.
Before giving the proof of this property, let us discuss the meaning of the additional conditions in the formulation. The condition a≠0 is necessary in order to avoid division by zero, since 0 n =0, and when we got acquainted with division, we agreed that it is impossible to divide by zero. The condition m>n is introduced so that we do not go beyond natural exponents. Indeed, for m>n the exponent a m−n is a natural number, otherwise it will be either zero (which happens for m−n ) or a negative number (which happens for m Proof. The main property of a fraction allows us to write the equality a m−n a n =a (m−n)+n =a m. From the obtained equality a m−n ·a n =a m and from it follows that a m−n is a quotient of powers of a m and a n . This proves the property of partial powers with the same bases. Let's take an example. Let's take two degrees with the same bases π and natural exponents 5 and 2, the considered property of the degree corresponds to the equality π 5: π 2 = π 5−3 = π 3. Now consider product degree property: the natural degree n of the product of any two real numbers a and b is equal to the product of the degrees a n and b n , that is, (a b) n =a n b n . Indeed, by definition of a degree with a natural exponent, we have Here's an example: This property extends to the degree of the product of three or more factors. That is, the natural power property n of the product of k factors is written as (a 1 a 2 ... a k) n =a 1 n a 2 n ... a k n. For clarity, we show this property with an example. For the product of three factors to the power of 7, we have . The next property is natural property: the quotient of the real numbers a and b , b≠0 to the natural power n is equal to the quotient of the powers a n and b n , that is, (a:b) n =a n:b n . The proof can be carried out using the previous property. So (a:b) n b n =((a:b) b) n =a n, and the equality (a:b) n b n =a n implies that (a:b) n is the quotient of a n divided by b n . Let's write this property using the example of specific numbers: Now let's voice exponentiation property: for any real number a and any natural numbers m and n, the power of a m to the power of n is equal to the power of a with exponent m·n , that is, (a m) n =a m·n . For example, (5 2) 3 =5 2 3 =5 6 . The proof of the power property in a degree is the following chain of equalities: The considered property can be extended to degree within degree within degree, and so on. For example, for any natural numbers p, q, r, and s, the equality It remains to dwell on the properties of comparing degrees with a natural exponent. We start by proving the comparison property of zero and power with a natural exponent. First, let's justify that a n >0 for any a>0 . The product of two positive numbers is a positive number, as follows from the definition of multiplication. This fact and the properties of multiplication allow us to assert that the result of multiplying any number of positive numbers will also be a positive number. And the power of a with natural exponent n is, by definition, the product of n factors, each of which is equal to a. These arguments allow us to assert that for any positive base a the degree of a n is a positive number. By virtue of the proved property 3 5 >0 , (0.00201) 2 >0 and It is quite obvious that for any natural n with a=0 the degree of a n is zero. Indeed, 0 n =0·0·…·0=0 . For example, 0 3 =0 and 0 762 =0 . Let's move on to negative bases. Let's start with the case when the exponent is an even number, denote it as 2 m , where m is a natural number. Then Finally, when the base of a is a negative number and the exponent is an odd number 2 m−1, then We turn to the property of comparing degrees with the same natural exponents, which has the following formulation: of two degrees with the same natural exponents, n is less than the one whose base is less, and more than the one whose base is greater. Let's prove it. Inequality a n properties of inequalities the inequality being proved of the form a n (2,2) 7 and It remains to prove the last of the listed properties of powers with natural exponents. Let's formulate it. Of the two degrees with natural indicators and the same positive bases, less than one, the degree is greater, the indicator of which is less; and of two degrees with natural indicators and the same bases greater than one, the degree whose indicator is greater is greater. We turn to the proof of this property. Let us prove that for m>n and 0 It remains to prove the second part of the property. Let us prove that for m>n and a>1, a m >a n is true. The difference a m −a n after taking a n out of brackets takes the form a n ·(a m−n −1) . This product is positive, since for a>1 the degree of an is a positive number, and the difference am−n−1 is a positive number, since m−n>0 by virtue of the initial condition, and for a>1 the degree of am−n is greater than one . Therefore, a m − a n >0 and a m >a n , which was to be proved. This property is illustrated by the inequality 3 7 >3 2 .
. The last product, based on the properties of multiplication, can be rewritten as 
, which is equal to a n b n .
.
.
.
. For greater clarity, here is an example with specific numbers: (((5,2) 3) 2) 5 =(5,2) 3+2+5 =(5,2) 10
.
.
. For each of the products of the form a·a is equal to the product of the modules of the numbers a and a, therefore, is a positive number. Therefore, the product will also be positive. 
and degree a 2 m . Here are examples: (−6) 4 >0 , (−2,2) 12 >0 and .
. All products a·a are positive numbers, the product of these positive numbers is also positive, and its multiplication by the remaining negative number a results in a negative number. Due to this property (−5) 3<0
, (−0,003) 17 <0
и 
.
.
Properties of degrees with integer exponents
Since positive integers are natural numbers, then all properties of powers with positive integer exponents exactly coincide with the properties of powers with natural exponents listed and proven in the previous paragraph.
The degree with a negative integer exponent, as well as the degree with a zero exponent, we defined in such a way that all properties of degrees with natural exponents expressed by equalities remain valid. Therefore, all these properties are valid both for zero exponents and for negative exponents, while, of course, the bases of the degrees are nonzero.
So, for any real and non-zero numbers a and b, as well as any integers m and n, the following are true properties of degrees with integer exponents:
- a m a n \u003d a m + n;
- a m: a n = a m−n ;
- (a b) n = a n b n ;
- (a:b) n =a n:b n ;
- (a m) n = a m n ;
- if n is a positive integer, a and b are positive numbers, and a b-n;
- if m and n are integers, and m>n , then at 0
For a=0, the powers a m and a n make sense only when both m and n are positive integers, that is, natural numbers. Thus, the properties just written are also valid for the cases when a=0 and the numbers m and n are positive integers.
It is not difficult to prove each of these properties, for this it is enough to use the definitions of the degree with a natural and integer exponent, as well as the properties of actions with real numbers. As an example, let's prove that the power property holds for both positive integers and nonpositive integers. To do this, we need to show that if p is zero or a natural number and q is zero or a natural number, then the equalities (ap) q =ap q , (a −p) q =a (−p) q , (ap ) −q =ap (−q) and (a−p)−q =a (−p) (−q). Let's do it.
For positive p and q, the equality (a p) q =a p·q was proved in the previous subsection. If p=0 , then we have (a 0) q =1 q =1 and a 0 q =a 0 =1 , whence (a 0) q =a 0 q . Similarly, if q=0 , then (a p) 0 =1 and a p 0 =a 0 =1 , whence (a p) 0 =a p 0 . If both p=0 and q=0 , then (a 0) 0 =1 0 =1 and a 0 0 =a 0 =1 , whence (a 0) 0 =a 0 0 .
Let us now prove that (a −p) q =a (−p) q . By definition of a degree with a negative integer exponent , then 
. By the property of the quotient in the degree, we have 
. Since 1 p =1·1·…·1=1 and , then . The last expression is, by definition, a power of the form a −(p q) , which, by virtue of the multiplication rules, can be written as a (−p) q .
Similarly 
.
AND 
.
By the same principle, one can prove all other properties of a degree with an integer exponent, written in the form of equalities.
In the penultimate of the properties written down, it is worth dwelling on the proof of the inequality a −n >b −n , which is true for any negative integer −n and any positive a and b for which the condition a . Since by condition a 0 . The product a n ·b n is also positive as the product of positive numbers a n and b n . Then the resulting fraction is positive as a quotient of positive numbers b n − a n and a n b n . Hence, whence a −n >b −n , which was to be proved.
The last property of degrees with integer exponents is proved in the same way as the analogous property of degrees with natural exponents.
Properties of powers with rational exponents
We defined the degree with a fractional exponent by extending the properties of a degree with an integer exponent to it. In other words, degrees with fractional exponents have the same properties as degrees with integer exponents. Namely:
The proof of the properties of degrees with fractional exponents is based on the definition of a degree with a fractional exponent, on and on the properties of a degree with an integer exponent. Let's give proof.
By definition of the degree with a fractional exponent and , then 
. The properties of the arithmetic root allow us to write the following equalities. Further, using the property of the degree with an integer exponent, we obtain , whence, by the definition of a degree with a fractional exponent, we have 
, and the exponent of the degree obtained can be converted as follows: . This completes the proof.
The second property of powers with fractional exponents is proved in exactly the same way: 
The rest of the equalities are proved by similar principles: 
We turn to the proof of the next property. Let us prove that for any positive a and b , a b p . We write the rational number p as m/n , where m is an integer and n is a natural number. Conditions p<0 и p>0 in this case will be equivalent to the conditions m<0 и m>0 respectively. For m>0 and a
Similarly, for m<0 имеем a m >b m , whence , that is, and a p >b p .
It remains to prove the last of the listed properties. Let us prove that for rational numbers p and q , p>q for 0 
and 
. And the definition of a degree with a rational exponent allows us to pass to the inequalities and, respectively. From this we draw the final conclusion: for p>q and 0
Properties of degrees with irrational exponents
From how a degree with an irrational exponent is defined, it can be concluded that it has all the properties of degrees with rational exponents. So for any a>0 , b>0 and irrational numbers p and q the following are true properties of degrees with irrational exponents:
- a p a q = a p + q ;
- a p:a q = a p−q ;
- (a b) p = a p b p ;
- (a:b) p =a p:b p ;
- (a p) q = a p q ;
- for any positive numbers a and b , a 0 the inequality a p b p ;
- for irrational numbers p and q , p>q at 0
From this we can conclude that powers with any real exponents p and q for a>0 have the same properties.
Bibliography.
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics Zh textbook for 5 cells. educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 7 cells. educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 9 cells. educational institutions.
- Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
- Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).
