Lesson topic: Degree with a natural indicator

Lesson type: lesson in generalization and systematization of knowledge

Lesson type: combined

Forms of work: individual, frontal, work in pairs

Equipment: computer, media product (presentation in the programMicrosoftOffice Power Point 2007); cards with assignments for independent work

Lesson objectives:

Educational : working out 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: contribute to the formation of skills to apply methods of generalization, comparison, highlighting the main thing, the development of a mathematical outlook, thinking, speech, attention and memory.

- educational: to contribute to the fostering of interest in mathematics, activity, organization, to form a positive motive for learning, the development of educational and cognitive skills

Explanatory note.

This lesson is conducted in a general education class with an average level of mathematical training. The main task of the lesson is to practice 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 examples of design solutions. Different types of work are used in the lesson, which relieves fatigue of children.

Lesson structure:

1. Organizing time.

2. Posting the topic, setting the goals of the lesson.

4. Oral work.

5. Systematization of basic knowledge.

6. Elements of health-saving technologies.

7. Test task execution

9. Lesson summary.

10. Homework.

During the classes:

I.Organizing time

Teacher: Hello guys! I am pleased to welcome you to our lesson today. Sit down. I hope that both success and joy await us in the lesson today. And we, working in a team, will show our talent.

Be attentive throughout the lesson. Think, ask, offer - since we will walk the road to truth together.

Open your notebooks and write down the number, cool work

II... Posting a topic, setting lesson goals

1) Lesson topic. Epigraph of the lesson.(Slide 2,3)

“Let someone try to erase from mathematics

degree, and he will see that without them you will not go far ”M.V. Lomonosov

2) Setting the goals of the lesson.

Teacher: So, in the lesson we will repeat, summarize and bring into the system the studied material. Your task is to show your knowledge of the properties of the 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) solve the anagram: (slide 4)

Nispete (degree)

Ktoreosis (cut)

Ovaniosne (base)

Kazapotel (indicator)

Monunieje (multiplication)

2) What is a natural exponent degree?(Slide 5)

(The power of the number a with a natural indicator n greater than 1 is the expression a n equal to the product n factors, each of which is equal to a a-base, n -indicator)

3) Read the expression, name the base and 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 Have stale job

1) verbal counting (slide8)

Teacher: Now let's check how you can apply these formulas when solving.

1) x 5 x 7 ; 2) a 4 and 0 ;

3) to 9 : to 7 ; 4) r n : r ;

5)5 5 2 ; 6) (- b )(- b ) 3 (- b );

7) with 4 : from; 8) 7 3 : 49;

9) at 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 : at

2) the game "Eliminate unnecessary" ((- 1) 2 ) (slide 9)

-1

Well done. Did a good job. Then we solve the following examples.

V Systematization of basic knowledge

1. Connect by lines the expressions corresponding to each other:(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.Sort the numbers in ascending order:(slide 11)

3 2 (-0.5) 3 (½) 3 35 0 (-10) 3

3. Execution of the task followed by self-test(slide 12)

• A1 represent the product as a degree:

a) a) x 5 x 4 ; b) 3 7 3 9 ; at 4) 3 (-4) 8 .

• A 2 simplify the expression:

a) x 3 x 7 x 8 ; b) 2 21 :2 19 2 3

• And 3 do the exponentiation:

a) (a 5 ) 3 ; b) (-c 7 ) 2

VI Elements of health-saving technologies (slide 13)

Physical education: repetition of the degree of numbers 2 and 3

Vii Test task (slide14)

The answers to the test are written on the board: 1 d 2 o 3b 4y 5 h 6a (extraction)

VIII Independent work on cards

On each desk, cards with an assignment for options, after completing the work, are submitted for verification

Option 1

1) Simplify expressions:

and) b)

in) d)

and) b)

in) d)

Option 2

1) Simplify expressions:

and) b)

in) d)

2) Find the value of the expression:

and) b)

in) d)

3) Show with the arrow what the value of the expression is equal to: zero, positive or negative number:

IX Lessons learned

P / p No.

Type of work

self-esteem

Teacher assessment

1

Anagram

2

Read the expression

3

rules

4

Verbal counting

5

Connect with lines

6

Arrange in ascending order

7

Self-Test Assignments

8

Test

9

Independent work on cards

X Homework

Test cards

A1. Find the meaning of the expression: .

algebra 7th grade

mathematic teacher

branch MBOUTSOSH # 1

in the village of Poletaevo I.P. Zueva

Poletaevo 2016

Topic: « Natural exponent grade properties»

TARGET

1. Repetition, generalization and systematization of the studied material on the topic "Properties of the degree with a natural indicator."
2. Testing students' knowledge of this topic.
3. Application of the acquired knowledge when performing various tasks.

TASKS

subject :

to repeat, summarize 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;

meta-subject:

develop an operational style of thinking; promote the acquisition of communication skills by students working together; activate them creative thinking; Pcontinue the formation of certain competencies of students, which will contribute to their effective socialization; self-education and self-education skills.

personal:

educate culture, promote the formation of personal qualities aimed at a benevolent, tolerant attitude towards each other, people, life; foster initiative and independence in activity; bring 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,screen for projection, board, handouts.

Software: Windows 7 OS: MS Office 2007 (application required -PowerPoint).

Preparatory stage:

presentation "Properties of the degree with a natural indicator";

handout;

grade sheet.

Structure

Organizing time... Setting the 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 the ZUN of students on this topic in an interesting and effective form.

Lesson organization The lesson is held in grade 7. The guys work in pairs, independently, the teacher acts as a consultant-observer.

During the classes

Organizing time:

Hello guys! Today we have an unusual game lesson. Each of you is given a great opportunity to express yourself, to show your knowledge. Perhaps during the lesson you will reveal hidden abilities in yourself that will be useful to you in the future.

Each of you has a grade sheet and cards on the table for completing tasks in them. Pick up the grade sheet, you need it so that you yourself assess your knowledge during the lesson. Sign it up.

So, I invite you to the lesson!

Guys, look at the screen and listen to the poem.

Slide number 1

Multiply and divide

To raise a degree to a degree ...

These properties are familiar to us.

And they are not new for a long time.

Five simple rules of these

Everyone in the class has already answered

But if you forgot the properties,

Consider an example you haven't solved!

And in order to live without troubles at school

I'll give you some practical advice:

Do you want to forget the rule?

Just try to memorize!

Answer the question:

1) What actions are mentioned in it?

2) What do you think we will talk about today in the lesson?

Thus, the topic of our tutorial:

"Properties of a natural exponent" (Slide 3).

Setting the goals and objectives of the lesson

In the lesson, we will repeat, generalize and bring into the system the studied material on the topic "Properties of the degree with a natural indicator"

Let's see how you learned how to multiply and divide powers with the same bases, as well as raise a power to a power

Updating basic knowledge. Systematization of theoretical material.

1) Oral work

Let's work orally

1) Formulate the properties of the degree with a natural exponent.

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; a 0.

2) Checking the theoretical part (Card number 1)

Now pick up card number 1 andfill the gaps

1) If the exponent is an even number, then the value of the degree is always _______________

2) If the exponent is an odd number, then the value of the degree coincides with the sign of ____.

3) Product of degreesa n a k \u003d a n + k
When multiplying degrees with the same bases, the base is ____________, and the exponents are ________.

4) Private degreesa n: a k \u003d a n - k
When dividing degrees with the same bases, you need a base _____, and from the index of the dividend ____________________________.

5) Exponentiation (a n) к \u003d a nk
When raising a degree to a degree, the base must be _______, and the exponents are ______.

Checking answers. (Slides 10-13)

Main part

3) And now we open notebooks, write down the number 28.01 14g, great work

Game "Clapperboard » (Slide 14)

Complete assignments in notebooks yourself

Follow the steps: a)x11 ∙ x ∙ x2 b)x14 : x5 c) (a4 ) 3 d) (-Za)2 .

Compare the value of the expression with zero: a) (- 5)7 , b) (- 6)18 ,

at 4)11 . ( -4) 8 d) (- 5) 18 ∙ (- 5) 6 , d) - (- 4)8 .

Calculate the value of the 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 do one hand clap.

Calculate the number of points and enter them on the score sheet.

4) Now let's do eye gymnastics, relieve stress, and continue working. We closely monitor the movement of objects

Begin! (Slide 15,16,17,18).

5) Now let's get down to the next type of our work. (Card2)

Write the answer as a degree with a base FROM and you will learn the name and surname of the great French mathematician who was the first to introduce the concept of the power of a number.

Guess the name of the scientist mathematician.

1.	FROM 5 ∙ С 3	6.	FROM 7 : FROM 5
2.	FROM 8 : FROM 6	7.	(FROM 4 ) 3 ∙ С
3,	(FROM 4 ) 3	8.	FROM 4 ∙ FROM 5 ∙ С 0
4.	FROM 5 ∙ С 3 : FROM 6	9.	FROM 16 : FROM 8
5.	FROM 14 ∙ С 8	10.	(FROM 3 ) 5

ABOUT answer: RENE DECART

R

Sh

M

YU

TO

H

AND

T

E

D

FROM 8

FROM 5

FROM 1

FROM 40

FROM 13

FROM 12

FROM 9

FROM 15

FROM 2

FROM 22

Now let's listen to the student's message about "Rene Descartes"

René Descartes was born on March 21, 1596 in the small town of La Gay in Touraine. The genus Descartes belonged to the ignorant bureaucratic nobility. Rene spent his childhood in Touraine. In 1612 Descartes graduated from high school. He spent eight and a half years in it. Descartes did not immediately find his place in life. A nobleman by birth, after graduating from college in La Flèche, he plunges headlong into the high life of Paris, then gives up everything for the sake of science. Descartes assigned mathematics a special place in his system, he considered its principles of establishing truth as 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 the unknown; a, b, c - for coefficients, for degrees. Descartes' interests are not limited to mathematics, but include mechanics, optics, 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 blackboard:

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) \u003d -125

2. Calculate the value of the expression:

(5-x) 2 -2x 3 + 3x 2 -4x + x-x 0

a) for x \u003d -1

b) for x \u003d 2 Independently

7) Pick up card number 3, do the test

Option 1

Option 2.

1. Perform power division 217 : 2 5 2 12 2 45 2. Write in the form of a power (x + y) (x + y) \u003d x 2 + y 2 (x + y) 2 2 (x + y) 3. Replace * degree so that the equality afive · * \u003d 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 meaning of a fraction

1. Perform division of degrees 99 : 9 7 9 16 9 63 2. Write it down as a power (x-y) (x-y) \u003d ... x 2 -y 2 (x-y) 2 2 (x-y) 3. Replace * degree so that the equalityb 9 * \u003d b 18 b 17 b 1 1 4. What is the value of the expression(from 6) 4? a) from 10 b) from 6 c) from 24 5. From the proposed options, choose the one that can replace * in the equality (*)3 = 5 24 5 21 6 find the meaning of a fraction

Check each other's work and put your teammates on the grade sheet.

Option 1	and	b	b	from	b	3
Option 2	and	b	from	from	and	4

Additional tasks for strong learners

Each assignment is assessed separately.

Find the value of an expression:

8) Now let's see the effectiveness of our lesson ( Slide 19)

To do this, completing the task, cross out the letters corresponding to the answers.

AOVSTLKRICHGNMO

Simplify the expression:

1.	С 4 ∙ С 3	5.	(FROM 2 ) 3 ∙ FROM 5
2.	(C 5) 3	6.	FROM 6 ∙ FROM 5 : FROM 10
3.	C 11: C 6	7.	(FROM 4 ) 3 ∙ С 2
4.	С 5 ∙ С 5: С

Cipher: AND - From 7 IN- From 15 G - FROM And - From 30 K - S 9 M - From 14 H - S 13 ABOUT - From 12 R - S 11 FROM - S 5 T - C 8 H - C 3

What word did you get? ANSWER: EXCELLENT! (Slide 20)

Summing up, grading, grading (Slide 21)

Let's summarize our lesson, how successfully we repeated, summarized and systematized knowledge on the topic "Properties of a degree with a natural indicator"

We take the grade sheets and calculate the total number of points and write them down in the final grade line

Stand up who scored 29-32 points: the score is excellent

25-28 points: assessment is good

20-24 points: assessment - satisfactory

Once again, I will check the correctness of the tasks on the cards, check your results with the points in the test sheet. I will put the marks in the journal

And for active work in the assessment lesson:

Guys, I ask you to evaluate your activities in the lesson. Mark on the mood sheet.

Grade sheet
Last name First name		Assessment
1.Theoretical part
2. Game "Clapperboard"
3. Test
4. "Code"
Additional part
Final grade:
Emotional assessment	About myself	About the lesson
Satisfied
Dissatisfied

Homework (Slide 22)

Create a crossword puzzle with the keyword DEGREE. In the next lesson, we will look at the most interesting works.

№ 567

List of sources used

1. Textbook "Algebra Grade 7".
2. Poem. http://yandex.ru/yandsearch
3. NOT. Shchurkov. Culture modern lesson... Moscow: Russian Pedagogical Agency, 1997.
4. A.V. Petrov. Methodological and methodological foundations of personality-developing computer education. Volgograd. Change, 2001.
5. A.S. Belkin. Success situation. How to create it. M .: "Education", 1991.
6. Informatics and Education №3. Operational thinking style, 2003

Lesson on the topic: "The degree and its properties."

The purpose of the lesson:

To summarize the knowledge of students on the topic: "Degree with a natural indicator."

To achieve from students a conscious understanding of the definition of the degree, properties, the ability to apply them.

To teach to apply knowledge, skill for tasks of various complexity.

Create conditions for the manifestation of independence, perseverance, mental activity, instill a love of mathematics.

Equipment: punched cards, cards, tests, tables.

The lesson is designed with the aim of systematizing and generalizing the knowledge of students about the properties of a degree with a natural indicator. Lesson material forms mathematical knowledge in children and develops interest in the subject, outlook in the historical aspect.

Progress.

Communication of the topic and purpose of the lesson.

Today we have a generalizing lesson on the topic "Degree with a natural indicator and its properties."

The task of our lesson is to repeat all the material covered and prepare for the test.

Homework check.

(Purpose: to check the assimilation of exponentiation, product and power).

№ 238 (b) # 220 (a; d) # 216.

There are 2 people behind the board with individual cards.

a 4 ∙ a 15 a 12 ∙ a 4 a 12: a 4 a 18: a 9 (a 2) 5 (a 4) 8 (a 2 b 3) 6 (a 6 bb 4) 3 a 0 a 0

Oral work.

(Purpose: to repeat the key points that reinforce the algorithm for multiplying and dividing powers, exponentiation).

Formulate the definition of the degree of a number with a natural exponent.

Follow the steps.

a ∙ a 3; a 4: a 2; (a 6) 2; (2a 3) 3; a 0.

At what value of x does equality hold.

5 6 ∙ 5 x \u003d 5 10 10 x: 10 2 \u003d 10 (a 4) x \u003d a 8 (a x b 2) \u003d a 35 b 10

Determine the sign of the expression without performing any calculations.

(-3) 5 , -19 2 , -(-15) 2 , (-8) 6 , - (-17) 7

Simplify.

and)
; b) (a 4) 6: (a 3) 3

Brainstorm.

( purpose : check basic knowledge students, degree properties).

Working with punched cards, for speed.

a 6: a 4; a 10: a 3 (a 2) 2; (a 3) 3; (a 4) 5; (a 0) 2.

(2a 2) 2; (-2a 3) 3; (3a 4) 2; (-2a 2 b) 4.

The task: Simplify the expression (we work in pairs, the class solves the task a, b, c, we check collectively).

(Purpose: working out the properties of the degree with a natural indicator.)

and)
; b)
; in)

6. Calculate:

and)
(collectively )

b)
(by yourself )

in)
(by yourself )

d)
(collectively )

e)
(by yourself ).

7 . Check yourself!

(Purpose: the development of elements of creative activity of students and the ability to control their actions).

Work with tests, 2 students at the board, self-test.

I - c.

Evaluate expressions.

║ - in.

Simplify expressions.

Calculate.

Evaluate expressions.

D / s home k / r (by cards).

Summing up the results of the lesson, assigning marks.

(Purpose: For students to see clearly the result of their work, develop cognitive interest).

Who first started studying a degree?

How to build a n ?

To the nth degree weand erect

It is necessary to multiply n time

If a n one - never

If more - then multiplyand on a,

i repeat, n times.

3) Can we raise the number to n degree, very fast?

If you take a micro calculator

Number a you will dial only once

And then the sign "multiplication" - also once,

You will press the sign "succeed" so many times

how many n without one will show us

And the answer is ready, no school penEVEN.

4) List the properties of the degree with a natural exponent.

We will put grades for the lesson after checking the work with punched cards, with tests, taking into account the answers of those students who answered during the lesson.

You did a good job today, thank you.

Literature:

1.A.G. Mordkovich Algebra-7 class.

2.Didactic materials -7th grade.

3. A.G. Mordkovich Tests - grade 7.

After the degree of the number has been determined, it is logical to talk about properties degree... 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 in solving examples.

Page navigation.

Properties of natural exponents

By definition of a degree with a natural exponent, the degree a n is the product of n factors, each of which is equal to a. Based on this definition, and also using real multiplication properties, one can obtain and justify the following natural grade properties:

1. the main property of the degree a m · a n \u003d a m + n, its generalization;
2. property of private degrees with the same bases a m: a n \u003d a m − n;
3. product degree property (a b) n \u003d a n b n, its extension;
4. property of the quotient in natural degree (a: b) n \u003d a n: b n;
5. raising a power to a power (a m) n \u003d a mn, its generalization (((a n 1) n 2)…) n k \u003d a n 1 n 2… n k;
6. comparing power to zero:
   • if a\u003e 0, then a n\u003e 0 for any natural n;
   • if a \u003d 0, then a n \u003d 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 ;
7. if a and b are positive numbers and a
8. if m and n are integerssuch that m\u003e n, then at 0 0 the inequality a m\u003e a n is true.

Note right away that all the equalities written down are identical subject to the specified conditions, and their right and left parts can be swapped. For example, the main property of the fraction a m a n \u003d a m + n for simplifying expressions often used as a m + n \u003d a m · a n.

Now let's look at each of them in detail.

Let's start with the property of a product of two degrees 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 \u003d a m + n is true.

Let us prove the main property of the degree. By definition of a degree with a natural exponent, the product of degrees 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 the number a with natural exponent m + n, that is, a m + n. This completes the proof.

Let's give an example that confirms the main property of the degree. Take degrees with the same bases 2 and natural degrees 2 and 3, according to the basic property of the degree, we can write the equality 2 2 · 2 3 \u003d 2 2 + 3 \u003d 2 5. Let us check its validity, for which we calculate the values \u200b\u200bof the expressions 2 2 · 2 3 and 2 5. Exponentiation, we have 2 2 2 3 \u003d (2 2) (2 2 2) \u003d 4 8 \u003d 32 and 2 5 \u003d 2 2 2 2 2 2 \u003d 32, since equal values \u200b\u200bare obtained, the equality 2 2 2 3 \u003d 2 5 is true, and it confirms the main property of the degree.

The main property of the 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 natural numbers n 1, n 2, ..., n k, the equality a n 1 a n 2… a n k \u003d a n 1 + n 2 +… + n k.

For example, (2.1) 3 (2.1) 3 (2.1) 4 (2.1) 7 \u003d (2,1) 3+3+4+7 =(2,1) 17 .

You can go to the next property of degrees with a natural exponent - property of private degrees with the same bases: for any nonzero real number a and arbitrary natural numbers m and n satisfying the condition m\u003e n, the equality a m is true: a n \u003d a m − n.

Before giving a proof of this property, let us discuss the meaning of additional conditions in the formulation. The condition a ≠ 0 is necessary in order to avoid division by zero, since 0 n \u003d 0, and when we got acquainted with division, we agreed that one cannot divide by zero. The condition m\u003e n is introduced so that we do not go beyond the natural exponents. Indeed, for m\u003e 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 when m

Evidence. The main property of a fraction allows us to write the equality a m − n a n \u003d a (m − n) + n \u003d a m... From the obtained equality a m − n · a n \u003d a m and from it follows that a m − n is a quotient of powers a m and a n. This proved the property of private degrees with the same bases.

Let's give an example. 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 \u003d π 5−3 \u003d 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 powers of a n and b n, that is, (a b) n \u003d a n b n.

Indeed, by definition of a degree with a natural exponent, we have ... Based on the properties of multiplication, the last product can be rewritten as , which is equal to a n · b n.

Let's give an example: .

This property applies to the degree of the product of three or more factors. That is, the property of the natural degree n of the product of k factors is written as (a 1 a 2… a k) n \u003d a 1 n a 2 n… a k n.

For clarity, we will show this property by an example. For the product of three factors to the power of 7, we have.

The next property is private property in kind: the quotient of the real numbers a and b, b ≠ 0 in the natural power n is equal to the quotient of the powers of a n and b n, that is, (a: b) n \u003d a n: b n.

The proof can be carried out using the previous property. So (a: b) n b n \u003d ((a: b) b) n \u003d a n, and from the equality (a: b) n · b n \u003d a n it follows that (a: b) n is the quotient of dividing a n 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 degree of a m to the power n is equal to the power of the number a with exponent m n, that is, (a m) n \u003d a m n.

For example, (5 2) 3 \u003d 5 2 3 \u003d 5 6.

The proof of the property of degree to degree is the following chain of equalities: .

The considered property can be extended to degree to degree to degree, etc. For example, for any natural numbers p, q, r and s, the equality ... For clarity, here's an example with specific numbers: (((5,2) 3) 2) 5 =(5,2) 3+2+5 =(5,2) 10 .

It remains to dwell on the properties of comparing degrees with natural exponents.

Let's start by proving the property of comparing zero and degree with natural exponent.

First, let us prove that a n\u003e 0 for any a\u003e 0.

The product of two positive numbers is a positive number, which 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 degree of a number a with natural exponent n, by definition, is the product of n factors, each of which is equal to a. This reasoning allows us to assert that for any positive base a, the degree a n is a positive number. By virtue of the proved property 3 5\u003e 0, (0.00201) 2\u003e 0 and .

It is quite obvious that for any natural n for a \u003d 0 the degree of a n is zero. Indeed, 0 n \u003d 0 · 0 ·… · 0 \u003d 0. For example, 0 3 \u003d 0 and 0 762 \u003d 0.

Moving on to negative bases of the degree.

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 ... For each of the products of the form a · a is equal to the product of the absolute values \u200b\u200bof the numbers a and a, therefore, is a positive number. Therefore, the product and the degree a 2 · m. Let us give examples: (−6) 4\u003e 0, (−2,2) 12\u003e 0 and.

Finally, when the base of the exponent a is negative and the exponent is an odd number 2 m − 1, then ... All products a · a are positive numbers, the product of these positive numbers is also positive, and multiplying it by the remaining negative number a results in a negative number. Due to this property (−5) 3<0 , (−0,003) 17 <0 и .

We turn to the property of comparing degrees with the same natural indicators, which has the following formulation: of two degrees with the same natural indicators, n is less than the one whose base is less, and the greater is the one whose base is greater. Let's prove it.

Inequality a n properties of inequalities the proved inequality of the form a n (2,2) 7 and .

It remains to prove the last of the listed properties of degrees with natural exponents. Let's formulate it. Of two degrees with natural indicators and the same positive bases less than one, the greater is the degree whose indicator is less; and of two degrees with natural indicators and the same bases, greater than one, the greater is the degree, the indicator of which is greater. We pass to the proof of this property.

Let us prove that for m\u003e n and 0 0 by virtue of the initial condition m\u003e n, whence it follows that for 0

It remains to prove the second part of the property. Let us prove that a m\u003e a n holds for m\u003e n and a\u003e 1. The difference a m - a n after placing a n outside the parentheses takes the form a n · (a m − n −1). This product is positive, since for a\u003e 1 the degree of an is a positive number, and the difference am − n −1 is a positive number, since m − n\u003e 0 due to the initial condition, and for a\u003e 1, the degree of am − n is greater than one ... Consequently, a m - a n\u003e 0 and a m\u003e a n, as required. This property is illustrated by the inequality 3 7\u003e 3 2.

Properties of degrees with integer exponents

Since positive integers are natural numbers, all properties of degrees with positive integer exponents exactly coincide with the properties of degrees with natural exponents listed and proved in the previous section.

The degree with an integer negative exponent, as well as a degree with a zero exponent, we determined so that all properties of degrees with natural exponents, expressed by equalities, remained true. Therefore, all these properties are valid for both zero exponents and negative exponents, while, of course, the bases of the exponents are nonzero.

So, for any real and nonzero numbers a and b, as well as any integers m and n, the following are true properties of powers with integer exponents:

1. a m a n \u003d a m + n;
2. a m: a n \u003d a m − n;
3. (a b) n \u003d a n b n;
4. (a: b) n \u003d a n: b n;
5. (a m) n \u003d a m n;
6. if n is a positive integer, a and b are positive numbers, and a b −n;
7. if m and n are integers, and m\u003e n, then at 0 1 the inequality a m\u003e a n holds.

For a \u003d 0, the degrees 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 \u003d 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 natural and integer exponents, as well as the properties of actions with real numbers. As an example, let us prove that the property of degree to degree holds for both positive integers and non-positive integers. To do this, it is necessary to show that if p is zero or a natural number and q is zero or a natural number, then the equalities (ap) q \u003d ap q, (a - p) q \u003d a (−p) q, (ap ) −q \u003d ap (−q) and (a −p) −q \u003d a (−p) (−q)... Let's do it.

For positive p and q, the equality (a p) q \u003d a p q was proved in the previous section. If p \u003d 0, then we have (a 0) q \u003d 1 q \u003d 1 and a 0 q \u003d a 0 \u003d 1, whence (a 0) q \u003d a 0 q. Similarly, if q \u003d 0, then (a p) 0 \u003d 1 and a p · 0 \u003d a 0 \u003d 1, whence (a p) 0 \u003d a p · 0. If both p \u003d 0 and q \u003d 0, then (a 0) 0 \u003d 1 0 \u003d 1 and a 0 0 \u003d a 0 \u003d 1, whence (a 0) 0 \u003d a 0 0.

Now let us prove that (a - p) q \u003d a (- p) q. By definition of a degree with an integer negative exponent, then ... By the property of the quotient in degree, we have ... Since 1 p \u003d 1 · 1 ·… · 1 \u003d 1 and, then. The last expression is by definition a power of the form a - (p q), which, due to the rules of multiplication, 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 written properties, it is worth dwelling on the proof of the inequality a - n\u003e 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\u003e b - n, as required.

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 degrees with rational exponents

We determined a degree with a fractional exponent by extending to it the properties of a degree with a whole exponent. In other words, fractional exponents have the same properties as 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. Here are the proofs.

By definition of a 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 a 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 obtained degree can be transformed as follows:. This completes the proof.

The second property of degrees with fractional exponents is proved in exactly the same way:

The remaining equalities are proved by similar principles:

We pass to the proof of the following 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. The conditions p<0 и p>0 in this case, the conditions m<0 и m>0 respectively. For m\u003e 0 and a

Similarly, for m<0 имеем a m >b m, whence, that is, and a p\u003e b p.

It remains to prove the last of the listed properties. Let us prove that for rational numbers p and q, p\u003e q for 0 0 - inequality a p\u003e a q. We can always bring the rational numbers p and q to a common denominator, let in this we get ordinary fractions and, where m 1 and m 2 are integers, and n is natural. In this case, the condition p\u003e q will correspond to the condition m 1\u003e m 2, which follows from. Then, by the property of comparing degrees with the same bases and natural exponents at 0 1 - inequality a m 1\u003e a m 2. These inequalities in the properties of the roots can be rewritten accordingly as and ... And the definition of the degree with a rational exponent allows you to go to inequalities and, respectively. Hence, we draw the final conclusion: for p\u003e q and 0 0 - inequality a p\u003e a q.

Properties of degrees with irrational exponents

From how a degree with an irrational exponent is defined, we can conclude that it has all the properties of a degree with a rational exponent. So for any a\u003e 0, b\u003e 0, and irrational numbers p and q, the following properties of degrees with irrational exponents:

1. a p a q \u003d a p + q;
2. a p: a q \u003d a p − q;
3. (a b) p \u003d a p b p;
4. (a: b) p \u003d a p: b p;
5. (a p) q \u003d a p q;
6. for any positive numbers a and b, a 0 the inequality a p b p;
7. for irrational numbers p and q, p\u003e q at 0 0 - inequality a p\u003e a q.

Hence, we can conclude that degrees with any real exponents p and q for a\u003e 0 have the same properties.

List of references.

• Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. MathematicsZh textbook for 5th grade. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for grade 7. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for grade 8. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 9th grade. educational institutions.
• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginning of analysis: Textbook for 10 - 11 grades of educational institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a guide for applicants to technical schools).

Preview:

MUNICIPAL BUDGETARY EDUCATIONAL INSTITUTION

SECONDARY EDUCATIONAL SCHOOL № 11

MUNICIPAL EDUCATION CITY - ANAPA RESORT

Nomination "Physics and Mathematics (Mathematics)"

Plan - a summary of the lesson on the topic:

7th grade

Developed by: Bykova E.A., mathematics teacher of the highest qualification category

Anapa, 2013

Public lesson in algebra in the 7th grade on the topic:

"Properties of a natural exponent"

Lesson objectives:

Educational: - practicing 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.

Educational: - education of cognitive activity, a sense of responsibility, a culture of communication, a culture of dialogue.

Developing: - development of visual memory, mathematically literate speech, logical thinking, conscious perception of educational material.

Tasks:

1. Subject: to 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.

2. Metasubject: to develop an operational style of thinking, to promote the acquisition of communication skills by students when working together, to activate their creative thinking; continue the formation of certain competencies of students, which will contribute to their effective socialization, skills of self-education and self-education

3. Personal: educate culture, promote the formation of personal qualities aimed at a benevolent, tolerant attitude towards people and life; foster initiative and independence in activity; bring to an understanding of the need for the topic under study for successful preparation for the state final certification.

Lesson type: generalizing lesson on the topic.

Lesson type: combined.

Lesson structure:

1. Organizational moment.

2. Communication of the topic, goals and objectives of the lesson.

3. Reproduction of what has been learned and its application in standard situations.

4. Transfer of acquired knowledge, their primary application in new or changed conditions, in order to form skills.

5.Elements of health-saving technologies.

6. Students' independent performance of assignments under the supervision of a teacher.

7. Summing up the results of the lesson and setting homework.

Equipment: multimedia projector, computer.

Microsoft Office Power Point 2007 Presentation(Attachment 1)

Lesson plan:

	Lesson stage		Time
	Organizing time.	Focus students on the lesson	1 min.
	Homework check	Error correction	3 min.
	Communication of the topic, goals and objectives of the lesson.	Setting lesson goals	1 min.
	Oral work. Repetition of the properties of the degree with a natural exponent.	Update basic knowledge	7 minutes
	Training exercises.	Form the skill of converting degrees with a natural indicator.	10 minutes.
	Physical culture break.	Application of health-saving technologies	2 minutes.
	Individual verification work by cards.	Error correction	12 minutes
	Lesson summary.	Summarize the theoretical information obtained in the lesson	2 minutes
	Homework setting.	Explain the content of the homework	2 minutes

Literature:

1. Algebra: textbook. for 7 cl. general education. institutions / Yu.N. Makarychev, N.G. Mindyuk and others; edited by S.A. Telyakovsky. - M .: Education, 2008.

2. Zvavich L.I., Kuznetsova L.V., Suvorova S.B. Didactic materials on algebra for grade 7. - M .: Education, 2009.

3. Collection test items for thematic and final control. Algebra Grade 7 / S.A. Pushkin, I.L. Gusev. - M .: "Intellect", 2013.

4. T.Yu.Dyumina, A.A. Makhonina, “Algebra. Lesson plans. ", - Volgograd:" Teacher ", 2013

During the classes

1. Organizational moment.

2. Checking homework

3. Topic of the lesson. Goals and objectives of the lesson.

Math friends

Absolutely everyone needs it.

Work hard in the lesson

And success awaits you for sure!

4. Oral work.

a) Repetition of the properties of the degree with a natural indicator. A table is given. In the left column, fill in the missing places, in the right - complete tasks.

The power of the number a with a natural indicatorp called ____________p ____________, each of which isand.	1. Present the work as a degree: and). (-8) * (-8) * (-8) * (-8) * (-8) *; b). (x-y) * (x-y) * (x-y) * (x-y) *; 2. Raise to the power: 3 4 ; (-0,2) 3 ; (2/3) 2 What is the base and exponent of the degrees recorded?
When multiplying degrees with the same bases, ___________ is left the same, and ___________ is added.	Follow the steps: a 4 * a 12; a 6 * a 9 * a; 3 2 * 3 3
When dividing degrees with the same bases, ___________ is left the same, and from __________ the numerator _________ __________ the denominator.	Follow the steps: a 12: a 4; n 9: n 3: n; 3 5 : 3 2
When raising a degree to a degree, _______________ is left unchanged, and __________ is multiplied.	Follow the steps: ; (m 3) 7; (k 4) 5; (4 2) 3
When raising to a power, the product is raised to this power _____________ ____________ and the results are multiplied.	Perform exponentiation: (-2 a 3 b 2) 5; (1 / 3p 2 q 3) 3
Power of number a , not equal to zero, with zero exponent is	Calculate: 3x 0 at x \u003d 2.6

b) Performing tasks on transforming expressions containing degrees, the student made the following mistakes:(writing on the board)

1) a) ; b) ;

in) ; d) ;

2) a) ; b) ;

in) ; d) ;

3) a) ; b) ;

in) .

What definitions, properties, rules does the student not know?

5. Training exercises.

№ 447 - on the blackboard and in notebooks with detailed commentary using the properties of degrees;

No. 450 (a, c) - on the board and in notebooks;

No. 445 - orally.

6. Physical minutes

We got up quickly, smiled

Higher and higher pulled up.

Well, straighten your shoulders,

Raise, lower.

Turn right, left,

Touch your hands with your knees.

Sat down, got up, sat down, got up

And they ran on the spot.

Youth learns with you

Develop both will and wit.

7. Individual test work.

Each student completes the tasks, they are accompanied by a key that uses the entire alphabet in order to exclude guessing the answers by letters. When correct decision Is the correct word.

Tasks for each row are individual.

P / p No.	Task 1 row	P / p No.	Quest 2 row	P / p No.	Quest 3 row
	m 3 * m 2 * m 8		a 4 * a 3 * a 2		a 4 * a * a 3 * a
	p 20: p 17		(2 4 ) 5 : (2 7 ) 2		(7x) 2
	c 5: c 0		3 * 3 2 * 3 0		p * p 2 * p 0
	(3a) 3		(2y) 5		c * c 3 * c
	m * m 5 * m 3 * m 0		(m 2) 4 * m		m * m 4 * (m 2) 2 * m 0
	2 14 : 2 8		(2 3 ) 2		(2 3 ) 7 : (2 5 ) 3
	(-x) 3 * x 4		(-x 3) * (- x) 4		X 3 * (-x) 4
	(p * p 3): p 5		(p 2 * p 5): p 4 * p 0		(p 2) 4: p 5
	3 7 * (3 2 ) 3 : 3 10		(3 5 ) 2 * 3 7 : 3 14		(3 4 ) 2 * (3 2 ) 3 : 3 11

Key

	32y 5				49x 2				27a 3
								m 13
81a 3		16a 4		10y 5	9y 7			32x 5	49y 3

The results of the work are highlighted on the slide for self-testing:

Maths

8. Lesson summary:

Summing up the results of the lesson, assigning marks.

- List the properties of the degree with a natural exponent.

We will put grades for the lesson after checking the work with the tests, taking into account the answers of those students who answered during the lesson.

Guess the crossword puzzle

Vertically:

1. He divides the dividend
2. Elementary figure on a plane
3. True equality
4. One followed by nine zeros
5. It is added to the like
6. Two to the power of three

Horizontally:

2. The number of sides in a triangle

4. Sum of monomials

5. Summarize

7. A segment connecting a point of a circle with its center

8. Has a numerator and denominator

9. Assignment at home:

The power of the number a with natural exponent n is called ____________ n ____________, each of which is equal to a. 1. Present the work in the form of a degree: a). (-8) * (-8) * (-8) * (-8) * (-8) *; b). (x-y) * (x-y) * (x-y) * (x-y) *; 2. Raise to the power: 3 4; (-0.2) 3; (2/3) 2 What is the base and exponent of the degrees recorded? When multiplying degrees with the same bases, ___________ is left the same, and ___________ is added. Follow the steps: a 4 * a 12; a 6 * a 9 * a; 3 2 * 3 3 When dividing degrees with the same bases, ___________ remain the same, and from __________ the numerator _________ __________ the denominator. Follow the steps: a 12: a 4; n 9: n 3: n; 3 5: 3 2 When raising a degree to a power, _______________ is left unchanged, and __________ is multiplied. Follow the steps:; (m 3) 7; (k 4) 5; (4 2) 3 When raising to a power, the product is raised to this power _____________ ____________ and the results are multiplied. Perform exponentiation: (-2 a 3 b 2) 5; (1 / 3p 2 q 3) 3 The degree of a number a, which is not equal to zero, with zero exponent is equal to Calculate: 3 x 0 at x \u003d 2.6 Repeat!

Brainstorm

We got up quickly, smiled, We pulled ourselves higher and higher. Well, straighten your shoulders, Raise, lower. Turn to the right, to the left, Touch your hands with your knees. They sat down, got up, sat down, got up, And ran on the spot. The youth learns with you To develop both will and ingenuity.

Individual test work # p / p Task 1 row # p / p Task 2 row # p / p Task 3 row 1 m 3 * m 2 * m 8 1 a 4 * a 3 * a 2 1 a 4 * a * a 3 * a 2 p 20: p 17 2 (2 4) 5: (2 7) 2 2 (7x) 2 3 c 5: c 0 3 3 * 3 2 * 3 0 3 p * p 2 * p 0 4 (3a ) 3 4 (2y) 5 4 c * c 3 * c 5 m * m 5 * m 3 * m 0 5 (m 2) 4 * m 5 m * m 4 * (m 2) 2 * m 0 6 2 14 : 2 8 6 (2 3) 2 6 (2 3) 7: (2 5) 3 7 (-x) 3 * x 4 7 (-x 3) * (- x) 4 7 -x 3 * (-x ) 4 8 (p * p 3): p 5 8 (p 2 * p 5): p 4 * p 0 8 (p 2) 4: p 5 9 3 7 * (3 2) 3: 3 10 9 (3 5) 2 * 3 7: 3 14 9 (3 4) 2 * (3 2) 3: 3 11

Check yourself! Key! A B C D E F G H I K m 9 32y 5 81 a 9 x 3 49x 2 m 5 p 4 c 5 27a 3 L M N O P Q R S T U F 64 3 4 p 3 27 2 5 x 7 p 6 m 3 m 13 a 8 X Y Z Z W S B B B Y E Y 81a 3 c 7 16a 4 25 10y 5 9y 7 -x 7 a 2 32x 5 49y 3 R x 5

maths

GUESS THE CROSSWORD Vertically: 1. It divides the dividend 2. An elementary figure on the plane 3. True equality 4. One with nine zeros 5. It is added with a similar 6. Two to the power of three Horizontally: 2. The number of sides in a triangle 4. Sum monomials 5. Sum up 7. A segment connecting a point of a circle with its center 8. Has a numerator and a denominator

Lesson summary Grading Assignment for home Answer questions p. 101, No. 450 (b, d), No. 534, No. 453.