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.

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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:

  1. the main property of the degree a m ·a n =a m+n , its generalization ;
  2. the property of partial powers with the same bases a m:a n =a m−n ;
  3. product degree property (a b) n =a n b n , its extension ;
  4. quotient property in kind (a:b) n =a n:b n ;
  5. exponentiation (a m) n =a m n , its generalization (((a n 1) n 2) ...) n k =a n 1 n 2 ... n k;
  6. 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 ;
  7. if a and b are positive numbers and a
  8. if m and n are natural numbers such that m>n , then at 0 0 the inequality a m >a n is true.

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 example, (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 . The last product, based on the properties of multiplication, can be rewritten as , which is equal to a n b n .

    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 . For greater clarity, here is 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 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 . 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 .

    Finally, when the base of a is a negative number 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 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 .

    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 0 due to the initial condition m>n , whence it follows that at 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 a n is a positive number, and the difference a m−n −1 is a positive number, since m−n>0 due to the initial condition, and for a>1, the degree of a m−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 .

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 proved in the previous paragraph.

We defined the degree with a negative integer exponent, as well as the degree with a zero exponent, so 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:

  1. a m a n \u003d 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. (a m) n = 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>n , then at 0 1 the inequality a m >a n is fulfilled.

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 (a p) q =a p q , (a − p) q =a (−p) q , (a p ) −q =a p (−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 recorded properties, 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 0 – inequality a p >a q . We can always reduce rational numbers p and q to a common denominator, let us get ordinary fractions and, where m 1 and m 2 are integers, and n is a natural number. In this case, the condition p>q will correspond to the condition m 1 >m 2, which follows from . Then, by the property of comparing powers with the same bases and natural exponents at 0 1 – inequality a m 1 >a m 2 . These inequalities in terms of the properties of the roots can be rewritten, respectively, as 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 0 – inequality a p >a q .

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:

  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. for any positive numbers a and b , a 0 the inequality a p b p ;
  7. for irrational numbers p and q , p>q at 0 0 – inequality a p >a q .

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).
Lesson topic: Degree with a natural indicator

Lesson 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. Various types of work are used in the lesson, which relieves children's fatigue.

Lesson structure:

  1. Organizing time.

  2. Message topics, setting goals for the lesson.

  4. oral work.

  5. Systematization of basic knowledge.

  6. Elements of health-saving technologies.

  7. Execution of a test task

  9. Lesson results.

  10. 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 -index)

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 Job

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 : y

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: .

algebra 7th grade

mathematic teacher

branch MBOUTSOSH №1

in the village of Poletaevo Zueva I.P.

Poletaevo 2016

Subject: « Properties of degree with natural exponent»

TARGET

  1. Repetition, generalization and systematization of the studied material on the topic "Properties of a degree with a natural indicator".
  2. Checking students' knowledge on the topic.
  3. 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 ∙C 3

6.

WITH 7 : WITH 5

2.

WITH 8 : WITH 6

7.

(WITH 4 ) 3 ∙C

3,

(WITH 4 ) 3

8.

WITH 4 ∙ WITH 5 ∙ C 0

4.

WITH 5 ∙C 3 : WITH 6

9.

WITH 16 : WITH 8

5.

WITH 14 ∙ C 8

10.

(WITH 3 ) 5

ABOUT 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 the social life of Paris, then abandons 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 ∙C 2

4.

C 5 ∙C 5 : C

Cipher: A - From 7 IN- From 15 G - WITH AND - From 30 TO - From 9 M - From 14 H - From 13 ABOUT - 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 Me

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

  1. Textbook "Algebra Grade 7".
  2. Poem. http://yandex.ru/yandsearch
  3. NOT. Shchurkov. The culture of the 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. situation of success. How to create it. M .: "Enlightenment", 1991.
  6. Computer science and education №3. Operational Thinking Style, 2003

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

The purpose of the lesson:

    Summarize students' knowledge 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 how to apply knowledge, skills for tasks of various complexity.

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

Equipment: punched cards, cards, tests, tables.

The lesson is designed to systematize and generalize the knowledge of students about the properties of a degree with a natural indicator. The material of the lesson forms mathematical knowledge in children and develops interest in the subject, horizons in the historical aspect.


Progress.

    Message about the topic and purpose of the lesson.

Today we have a general 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.

    Checking homework.

(Goal: to check the mastery of exponentiation, product and degree).

№ 238(b) No. 220 (a; d) No. 216.

Behind the board are 2 people 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 (а 6 bв 4) 3 a 0 a 0

    oral work.

(Goal: 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.

    Take action.

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

    At what value of x does the equation hold?

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

    Determine the sign of the expression without performing calculations.

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

    Simplify.

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

    Brainstorm.

( Target : check the basic knowledge of students, the properties of the degree).

Work with punched cards, for speed.

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

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

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

A)
; b)
; V)


6. Calculate:

A)
( collectively )

b)
( on one's own )

V)
( on one's own )

G)
( collectively )

e)
( on one's own ).


7 . Check yourself!

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

Work with tests, 2 students at the blackboard, self-examination.

I - c.



    Compute expressions.



║ - V.

    Simplify expressions.


    Calculate.


    Compute expressions.


    D / s home to / r (on cards).

    Summing up the lesson, grading.

(Goal: So that students can visually see the result of their work, develop cognitive interest).

    Who first began to study the degree?

    How to raise a n ?

So that to the nth degree weA erect

We need to multiply n once

If n one - never

If more, then multiply a on a,

I repeat n times.

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

If you take a calculator

Number a you only get it once

And then the sign of "multiplication" - also once,

You will press the sign "it will turn out" so many times

How many n without unit will show us

And the answer is ready, without a school pen EVEN .

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

Grades for the lesson will be set 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 - Grade 7.

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