Comparison of degrees with a real indicator. Degree with a natural indicator and its properties

Independent work of a 1st year student on the topic Degrees with a valid indicator. Degree properties with real exponent (6 hours)

    Study theoretical material and make notes (2 hours)

    Solve the crossword puzzle (2 hours)

    Do homework (2 hours)

Reference and didactic material is provided below.

On the concept of degree with a rational exponent

Some of the mostcommon

Types of transcendental functions before

Totally indicative, open access to

Many studies.

L. Eiler

From the practice of solving increasingly complex algebraic problems and operating with powers, it became necessary to generalize the concept of degree and expand it by introducing zero, negative and fractional numbers as an exponent.

The equality a 0 = 1 (for ) was used in his writings at the beginning of the 15th century. Samarkand scientist al-Kashi. Regardless of him, the zero indicator was introduced by N. Shuke in the 15th century. The latter also introduced negative exponents. The idea of ​​fractional exponents is contained in the French mathematician N. Orem (XIV century) in his

work "Algorism of proportions". Instead of our sign, he wrote , instead he wrote 4. Orem verbally formulates the rules for actions with degrees, for example (in modern notation): , etc.

Later, fractional, as well as negative, exponents are found in "Complete Arithmetic" (1544) by the German mathematician M. Stiefel and S. Stevin. The latter writes that the root of the degree P from the number a can be counted as a degree a with a fraction.

The expediency of introducing zero, negative and fractional indicators and modern symbols was first written in detail in 1665 by the English mathematician John Vallis. His work was completed by I. Newton, who began to systematically apply new symbols, after which they entered into common use.

The introduction of a degree with a rational exponent is one of many examples of a generalization of the concept of a mathematical action. A degree with zero, negative, and fractional exponents is defined in such a way that the same rules of action apply to it as for a degree with a natural exponent, i.e., so that the basic properties of the originally defined concept of degree are preserved, namely:

The new definition of a degree with a rational exponent does not contradict the old definition of a degree with a natural exponent, i.e., the meaning of the new definition of a degree with a rational exponent is preserved for the particular case of a degree with a natural exponent. This principle, observed in the generalization of mathematical concepts, is called the principle of permanence (preservation, constancy). It was expressed in an imperfect form in 1830 by the English mathematician J. Peacock, and it was fully and clearly established by the German mathematician G. Hankel in 1867. The principle of permanence is also observed when generalizing the concept of a number and expanding it to the concept of a real number, and before that introduction of the concept of multiplication by a fraction, etc.

Power function andgraphicsolving equations andinequalities

Thanks to the discovery of the method of coordinates and analytical geometry, start from the 17th century. the generally applicable graphical study of functions and the graphical solution of equations became possible.

Power a function is a function of the form

where α is a constant real number. Initially, however, we restrict ourselves to rational values ​​of α and instead of equality (1) we write:

where - rational number. For and by definition, respectively, we have:

at=1, y = x.

schedule the first of these functions on the plane is a straight line parallel to the axis Oh, and the second is the bisector of the 1st and 3rd coordinate angles.

When the function graph is a parabola . Descartes, who denoted the first unknown by z, the second - through y, third - through x:, wrote the parabola equation like this: ( z- abscissa). He often used a parabola to solve equations. To solve, for example, a 4th degree equation

Descartes via substitution

got a quadratic equation with two unknowns:

depicting a circle located in one plane (zx) with parabola (4). Thus, Descartes, introducing the second unknown (X), splits equation (3) into two equations (4) and (5), each of which represents a certain locus of points. The ordinates of their intersection points give the roots of equation (3).

“One day the king decided to choose his first assistant from among his courtiers. He led everyone to a huge castle. "Whoever opens it first will be the first helper." No one even touched the castle. Only one vizier came up and pushed the lock, which opened. It was not locked.

Then the king said: “You will receive this position because you rely not only on what you see and hear, but rely on your own strength and are not afraid to make an attempt.”

And today we will try, try to come to the right decision.

1. What mathematical concept are the words associated with:

Base

Indicator (Degree)

What words can combine the words:

rational number

Integer

Natural number

Irrational number (Real number)

Formulate the topic of the lesson. (Power with real exponent)

- repeat the properties of the degree

– consider the use of degree properties in calculations and simplifications of expressions

- development of computational skills.

So, a p, where p is a real number.

Give examples (choose from expressions 5–2, , 43, ) degrees

- with a natural indicator

- with integer value

- with a rational indicator

- with an irrational indicator

For what values ​​of a does the expression make sense?

a n , where n (a is any)

a m , where m (and not equal to 0) How to go from a negative exponent to a positive exponent?

Where p, q (a > 0)

What actions (mathematical operations) can be performed with degrees?

Set match:

When multiplying powers with equal bases

The bases are multiplied, but the exponent remains the same

When dividing powers with equal bases

The bases are divided, but the exponent remains the same


After it has been determined degree of, 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 determining the degree with a natural indicator 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 . Fulfilling 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 statement. 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 proven in the previous paragraph.

Degree with integer negative exponent, as well as the degree with 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:

  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 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

Degree with a fractional exponent we determined by extending to it the properties of a degree with an integer exponent. 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 the 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 it is defined degree with an irrational exponent, we can conclude that it has all the properties of powers 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).

We remind you that in this lesson we understand degree properties with natural indicators and zero. Degrees with rational indicators and their properties will be discussed in lessons for grade 8.

An exponent with a natural exponent has several important properties that allow you to simplify calculations in exponent examples.

Property #1
Product of powers

Remember!

When multiplying powers with the same base, the base remains unchanged, and the exponents are added.

a m a n \u003d a m + n, where " a"- any number, and" m", " n"- any natural numbers.

This property of powers also affects the product of three or more powers.

  • Simplify the expression.
    b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15
  • Present as a degree.
    6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17
  • Present as a degree.
    (0.8) 3 (0.8) 12 = (0.8) 3 + 12 = (0.8) 15

Important!

Please note that in the indicated property it was only about multiplying powers with the same grounds . It does not apply to their addition.

You cannot replace the sum (3 3 + 3 2) with 3 5 . This is understandable if
calculate (3 3 + 3 2) = (27 + 9) = 36 and 3 5 = 243

Property #2
Private degrees

Remember!

When dividing powers with the same base, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.

= 11 3 − 2 4 2 − 1 = 11 4 = 44
  • Example. Solve the equation. We use the property of partial degrees.
    3 8: t = 3 4

    T = 3 8 − 4

     Answer: t = 3 4 = 81

    • Using properties No. 1 and No. 2, you can easily simplify expressions and perform calculations.

    • Example. Simplify the expression.
      4 5m + 6 4 m + 2: 4 4m + 3 = 4 5m + 6 + m + 2: 4 4m + 3 = 4 6m + 8 − 4m − 3 = 4 2m + 5
    • Example. Find the value of an expression using degree properties.
      = = = 2 9 + 2
      2 5
      = 2 11
      2 5
      = 2 11 − 5 = 2 6 = 64

      Important!

      Please note that property 2 dealt only with the division of powers with the same bases.

      You cannot replace the difference (4 3 −4 2) with 4 1 . This is understandable if we consider (4 3 −4 2) = (64 − 16) = 48 , and 4 1 = 4

      Be careful!

      Property #3
      Exponentiation

      Remember!

      When raising a power to a power, the base of the power remains unchanged, and the exponents are multiplied.

      (a n) m \u003d a n m, where "a" is any number, and "m", "n" are any natural numbers.


      Properties 4
      Product degree

      Remember!

      When raising a product to a power, each of the factors is raised to a power. The results are then multiplied.

      (a b) n \u003d a n b n, where "a", "b" are any rational numbers; "n" - any natural number.

      • Example 1
        (6 a 2 b 3 c) 2 = 6 2 a 2 2 b 3 2 s 1 2 = 36 a 4 b 6 s 2
      • Example 2
        (−x 2 y) 6 = ((−1) 6 x 2 6 y 1 6) = x 12 y 6

      Important!

      Please note that property No. 4, like other properties of degrees, is also applied in reverse order.

      (a n b n)= (a b) n

      That is, to multiply degrees with the same exponents, you can multiply the bases, and leave the exponent unchanged.

      • Example. Calculate.
        2 4 5 4 = (2 5) 4 = 10 4 = 10,000
      • Example. Calculate.
        0.5 16 2 16 = (0.5 2) 16 = 1

      In more complex examples, there may be cases when multiplication and division must be performed on powers with different bases and different exponents. In this case, we advise you to do the following.

      For example, 4 5 3 2 = 4 3 4 2 3 2 = 4 3 (4 3) 2 = 64 12 2 = 64 144 = 9216

      Example of exponentiation of a decimal fraction.

      4 21 (−0.25) 20 = 4 4 20 (−0.25) 20 = 4 (4 (−0.25)) 20 = 4 (−1) 20 = 4 1 = 4

      Properties 5
      Power of the quotient (fractions)

      Remember!

      To raise a quotient to a power, you can raise the dividend and divisor separately to this power, and divide the first result by the second.

      (a: b) n \u003d a n: b n, where "a", "b" are any rational numbers, b ≠ 0, n is any natural number.

      • Example. Express the expression as partial powers.
        (5: 3) 12 = 5 12: 3 12

      We remind you that a quotient can be represented as a fraction. Therefore, we will dwell on the topic of raising a fraction to a power in more detail on the next page.

    This lesson is part of the topic "Conversions of expressions containing powers and roots".

    The summary is a detailed development of a lesson on the properties of a degree with a rational and real indicator. Computer, group and game learning technologies are used.

    Download:


    Preview:

    Methodical development of a lesson in algebra

    teacher of mathematics GAU KO PO KST

    Pekhova Nadezhda Yurievna

    on the topic: "Properties of a degree with a rational and real exponent."

    Lesson Objectives:

    • educational: consolidation and deepening of knowledge of the properties of a degree with a rational indicator and their application in exercises; improving knowledge on the history of the development of degrees;
    • developing: development of the skill of self- and mutual control; development of intellectual abilities, thinking skills,
    • educating: education of cognitive interest in the subject, education of responsibility for the work performed, to help create an atmosphere of active creative work.

    Lesson type: Lessons for improving knowledge, skills and abilities.

    Methods of conducting: verbal - visual.

    Pedagogical technologies: computer, group and game learning technologies.

    Lesson equipment: projection equipment, computer, presentation for the lesson, working

    notebooks, textbooks, cards with the text of a crossword puzzle and a reflective test.

    Lesson time: 1 hour 20 min.

    The main stages of the lesson:

    1. Organizational moment. Message topics, objectives of the lesson.

    2. Actualization of basic knowledge. Repetition of the properties of a degree with a rational exponent.

    3. Mathematical dictation on the properties of a degree with a rational exponent.

    4. Students' messages using a computer presentation.

    5. Work in groups.

    6. Crossword solution.

    7. Summing up, grading. Reflection.

    8. Homework.

    During the classes :

    1. Org. moment. Message about the topic, lesson objectives, lesson plan. Slides 1, 2.

    2. Updating of basic knowledge.

    1) Repetition of the properties of the degree with a rational indicator: students must continue the written properties - a frontal survey. Slide 3.

    2) Students at the blackboard - analysis of exercises from the textbook (Alimov Sh.A.): a) No. 74, b) No. 77.

    C) No. 82-a; b; c.

    No. 74: a) = = a;

    B) + = ;

    B) : = = = b.

    No. 77: a) = = ;

    B) = = = b.

    No. 82: a) = = = ;

    B) = = y;

    B) () () = .

    3. Mathematical dictation with mutual verification. Students share their work, compare answers and give grades.

    Slides 4 - 5

    4. Messages of students of some historical facts on the topic under study.

    Slides 6 - 12:

    First student: Slide 6

    The concept of a degree with a natural indicator was formed even among the ancient peoples. square and cubenumbers were used to calculate areas and volumes. The powers of some numbers were used in solving certain problems by scientists of ancient Egypt and Babylon.

    In the 3rd century, a book by the Greek scholar Diophantus was published"Arithmetic", in which the introduction of alphabetic symbols was initiated. Diophantus introduces symbols for the first six powers of the unknown and their reciprocals. In this book, a square is denoted by a sign and an index; for example, a cube is sign k with index r, etc.

    Second student: Slide 7

    A great contribution to the development of the concept of degree was made by the ancient Greek scientist Pythagoras. He had a whole school, and all his students were called Pythagoreans. They came up with the idea that each number can be represented in the form of figures. For example, they represented the numbers 4, 9 and 16 as squares.

    First student: Slides 8-9

    Slide 8

    Slide 9

    XVI century. In this century, the concept of degree has expanded: it began to be attributed not only to a specific number, but also to a variable. As they said then "to numbers in general" English mathematician S. Stevin coined a notation to denote the degree: notation 3(3)+5(2)–4 denoted such a modern notation 3 3 + 5 2 – 4.

    Second student: Slide 10

    Later, fractional and negative exponents are found in “Complete Arithmetic” (1544) by the German mathematician M. Stiefel and S. Stevin.

    S. Stevin suggested to mean by degree with an indicator of the form root, i.e. .

    First student: Slide 11

    At the end of the sixteenth century, Francois Vietintroduced letters to denote not only variables, but also their coefficients. He used abbreviations: N, Q, C - for the first, second and third degrees.

    But modern designations (such as, ) was introduced by René Descartes in the 17th century.

    Second student: Slide 12

    Modern definitionsand notation of degree with zero, negative and fractional exponent originate from the work of English mathematicians John Wallis (1616-1703) and Isaac Newton.

    5. Crossword solution.

    Students are given crossword puzzles. Solve in pairs. The pair that decides first gets the score. Slides 13-15.

    6. Group work. slide 16.

    Students perform independent work, working in groups of 4, advising each other. The work is then submitted for review.

    7. Summing up, grading.

    Reflection.

    Students complete a reflective test. Mark "+" if you agree, and "-" otherwise.

    Reflective test:

    1. I learned a lot of new things.

    2. It will be useful to me in the future.

    3. There was something to think about in the lesson.

    4. I received (a) answers to all the questions that arose during the lesson.

    5. At the lesson, I worked conscientiously and achieved the objectives of the lesson.

    8. Homework: Slide 17.

    1) № 76 (1; 3); № 70 (1; 2)

    2) Optional: make a crossword puzzle with the main concepts of the topic studied.

    References:

    1. Alimov Sh.A. algebra and the beginning of analysis grades 10-11, textbook - M .: Education, 2010.
    2. Algebra and the beginning of analysis Grade 10. Didactic materials. Enlightenment, 2012.

    Internet resources:

    1. Educational site - RusCopyBook.Com - Electronic textbooks and GDZ
    2. Site Educational resources of the Internet - for schoolchildren and students. http://www.alleng.ru/edu/educ.htm
    3. Website Teacher Portal - http://www.uchportal.ru/

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