Degree with rational exponent option 3. Properties of degrees, formulations, proofs, examples

MBOU "Sidorskaya"

comprehensive school»

Development of an outline plan open lesson

in algebra in 11th grade on the topic:

Prepared and carried out

math teacher

Iskhakova E.F.

Outline of an open lesson in algebra in 11th grade.

Subject : “A degree with a rational exponent.”

Lesson type : Learning new material

Lesson Objectives:

Introduce students to the concept of a degree with a rational exponent and its basic properties, based on previously studied material (degree with an integer exponent).

Develop computational skills and the ability to convert and compare numbers with rational exponents.

To develop mathematical literacy and mathematical interest in students.

Equipment : Task cards, student presentation by degree with an integer indicator, teacher presentation by degree with a rational indicator, laptop, multimedia projector, screen.

During the classes:

Organizing time.

Checking the mastery of the covered topic using individual task cards.

Task No. 1.

=2;

B) =x + 5;

Solve the system irrational equations: - 3 = -10,

4 - 5 =6.

Task No. 2.

Solve the irrational equation: = - 3;

B) = x - 2;

Solve the system of irrational equations: 2 + = 8,

3 - 2 = - 2.

Communicate the topic and objectives of the lesson.

The topic of our lesson today is “ Power with rational exponent».

Explanation of new material using the example of previously studied material.

You are already familiar with the concept of a degree with an integer exponent. Who will help me remember them?

Repetition using presentation " Degree with an integer exponent».

For any numbers a, b and any integers m and n the equalities are true:

a m * a n =a m+n ;

a m: a n =a m-n (a ≠ 0);

(a m) n = a mn ;

(a b) n =a n * b n ;

(a/b) n = a n /b n (b ≠ 0) ;

a 1 =a ;

a 0 = 1(a ≠ 0) Today we will generalize the concept of power of a number and give meaning to expressions that have a fractional exponent. Let's introduce definition

degrees with a rational exponent (Presentation “Degree with a rational exponent”): > Power of a 0 with rational exponent = r , Where m is an integer, and n is an integer, and > – natural ( , Where .

1), called the number = So, by definition we get that .

Let's try to apply this definition when completing a task.

EXAMPLE No. 1

I Present the expression as a root of a number:

A) B) IN) .

Now let's try to apply this definition in reverse

II Express the expression as a power with a rational exponent:

A) 2 B) IN) 5 .

The power of 0 is defined only for positive exponents.

0 r= 0 for any r> 0.

Using this definition, Houses you will complete #428 and #429.

Let us now show that with the definition of a degree with a rational exponent formulated above, the basic properties of degrees are preserved, which are true for any exponents.

For any rational numbers r and s and any positive a and b, the equalities are true:

1 0 . a r a s =a r+s ;

EXAMPLE: *

20 . a r: a s =a r-s ;

EXAMPLE: :

3 0 . (a r ) s =a rs ;

EXAMPLE: ( -2/3

4 0 . ( ab) r = a r b r ; 5 0 . ( = .

EXAMPLE: (25 4) 1/2 ; ( ) 1/2

EXAMPLE of using several properties at once: * : .

Physical education minute.

We put the pens on the desk, straightened the backs, and now we reach forward, we want to touch the board. Now we’ve raised it and leaned right, left, forward, back. You showed me your hands, now show me how your fingers can dance.

Working on the material

Let us note two more properties of degrees with rational exponents:

6 0 . Let r is a rational number and 0< a < b . Тогда

a r < b r at r> 0,

a r < b r at r< 0.

7 0 . For any rational numbersr And s from inequality r> s follows that

a r>a r for a > 1,

a r < а r at 0< а < 1.

EXAMPLE: Compare the numbers:

AND ; 2 300 and 3 200 .

Lesson summary:

Today in the lesson we recalled the properties of a degree with an integer exponent, learned the definition and basic properties of a degree with a rational exponent, and examined the application of this theoretical material in practice when performing exercises. I would like to draw your attention to the fact that the topic “Exponent with a rational exponent” is mandatory in Unified State Exam assignments. When preparing homework ( No. 428 and No. 429

In this article we will figure out what it is degree of. Here we will give definitions of the power of a number, while we will consider in detail all possible exponents, starting with the natural exponent and ending with the irrational one. In the material you will find a lot of examples of degrees, covering all the subtleties that arise.

Page navigation.

Power with natural exponent, square of a number, cube of a number

Let's start with . Looking ahead, let's say that the definition of the power of a number a with natural exponent n is given for a, which we will call degree basis, and n, which we will call exponent. We also note that a degree with a natural exponent is determined through a product, so to understand the material below you need to have an understanding of multiplying numbers.

Definition.

Power of a number with natural exponent n is an expression of the form a n, the value of which is equal to the product of n factors, each of which is equal to a, that is, .
In particular, the power of a number a with exponent 1 is the number a itself, that is, a 1 =a.

It’s worth mentioning right away about the rules for reading degrees. Universal method reading the entry a n is: “a to the power of n”. In some cases, the following options are also acceptable: “a to the nth power” and “nth power of a”. For example, let's take the power 8 12, this is “eight to the power of twelve”, or “eight to the twelfth power”, or “twelfth power of eight”.

The second power of a number, as well as the third power of a number, have their own names. The second power of a number is called square the number, for example, 7 2 is read as “seven squared” or “the square of the number seven.” The third power of a number is called cubed numbers, for example, 5 3 can be read as “five cubed” or you can say “cube of the number 5”.

It's time to bring examples of degrees with natural exponents. Let's start with the degree 5 7, here 5 is the base of the degree, and 7 is the exponent. Let's give another example: 4.32 is the base, and natural number 9 – exponent (4.32) 9 .

Please note that in the last example, the base of the power 4.32 is written in parentheses: to avoid discrepancies, we will put in parentheses all bases of the power that are different from natural numbers. As an example, we give the following degrees with natural exponents , their bases are not natural numbers, so they are written in parentheses. Well, for complete clarity, at this point we will show the difference contained in records of the form (−2) 3 and −2 3. The expression (−2) 3 is a power of −2 with a natural exponent of 3, and the expression −2 3 (it can be written as −(2 3) ) corresponds to the number, the value of the power 2 3 .

Note that there is a notation for the power of a number a with an exponent n of the form a^n. Moreover, if n is a multi-valued natural number, then the exponent is taken in brackets. For example, 4^9 is another notation for the power of 4 9 . And here are some more examples of writing degrees using the symbol “^”: 14^(21) , (−2,1)^(155) . In what follows, we will primarily use degree notation of the form a n .

One of the problems inverse to raising to a power with a natural exponent is the problem of finding the base of a power from a known value of the power and a known exponent. This task leads to .

It is known that the set of rational numbers consists of integers and fractions, and each a fractional number can be represented as positive or negative common fraction. We defined a degree with an integer exponent in the previous paragraph, therefore, in order to complete the definition of a degree with a rational exponent, we need to give meaning to the degree of the number a with a fractional exponent m/n, where m is an integer and n is a natural number. Let's do it.

Let's consider a degree with a fractional exponent of the form . For the power-to-power property to remain valid, the equality must hold . If we take into account the resulting equality and how we determined , then it is logical to accept it provided that for given m, n and a the expression makes sense.

It is easy to check that for all properties of a degree with an integer exponent are valid (this was done in the section properties of a degree with a rational exponent).

The above reasoning allows us to make the following conclusion: if given m, n and a the expression makes sense, then the power of a with a fractional exponent m/n is called the nth root of a to the power of m.

This statement brings us close to the definition of a degree with a fractional exponent. All that remains is to describe at what m, n and a the expression makes sense. Depending on the restrictions placed on m, n and a, there are two main approaches.

The easiest way is to impose a constraint on a by taking a≥0 for positive m and a>0 for negative m (since for m≤0 the degree 0 of m is not defined). Then we get the following definition of a degree with a fractional exponent.

Definition.

Power of a positive number a with fractional exponent m/n, where m is an integer and n is a natural number, is called the nth root of the number a to the power of m, that is, .

The fractional power of zero is also determined with the only caveat that the indicator must be positive.

Definition.

Power of zero with fractional positive exponent m/n, where m is a positive integer and n is a natural number, is defined as .
When the degree is not determined, that is, the degree of the number zero with a fractional negative exponent does not make sense.

It should be noted that with this definition of a degree with a fractional exponent, there is one caveat: for some negative a and some m and n, the expression makes sense, and we discarded these cases by introducing the condition a≥0. For example, the entries make sense or , and the definition given above forces us to say that powers with a fractional exponent of the form do not make sense, since the base should not be negative.

Another approach to determining a degree with a fractional exponent m/n is to separately consider even and odd exponents of the root. This approach requires an additional condition: the power of the number a, the exponent of which is , is considered to be the power of the number a, the exponent of which is the corresponding irreducible fraction (we will explain the importance of this condition below). That is, if m/n is an irreducible fraction, then for any natural number k the degree is first replaced by .

For even n and positive m, the expression makes sense for any non-negative a (even root of negative number does not make sense), for negative m the number a must still be different from zero (otherwise there will be division by zero). And for odd n and positive m, the number a can be any (the root of an odd degree is defined for any real number), and for negative m, the number a must be different from zero (so that there is no division by zero).

The above reasoning leads us to this definition of a degree with a fractional exponent.

Definition.

Let m/n be an irreducible fraction, m an integer, and n a natural number. For any reducible fraction, the degree is replaced by . The power of a number with an irreducible fractional exponent m/n is for



Let us explain why a degree with a reducible fractional exponent is first replaced by a degree with an irreducible exponent. If we simply defined the degree as , and did not make a reservation about the irreducibility of the fraction m/n, then we would be faced with situations similar to the following: since 6/10 = 3/5, then the equality must hold , But , A .

After the degree of a number has been determined, it is logical to talk about degree properties. In this article we will give the basic properties of the power of a number, while touching on all possible exponents. Here we will provide proofs of all properties of degrees, and also show how these properties are used when solving examples.

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Properties of degrees with natural exponents

By definition of a power with a natural exponent, the power a n is the product of n factors, each of which is equal to a. Based on this definition, and also using properties of multiplication of real numbers, 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. property of quotient powers with on the same grounds a m:a n =a m−n ;
3. product power property (a·b) n =a n ·b n , its extension;
4. property of the quotient to the natural degree (a:b) n =a n:b n ;
5. raising a degree to a power (a m) n =a m·n, its generalization (((a n 1) n 2) …) n k =a n 1 ·n 2 ·…·n k;
6. comparison of degree with zero:
   • if a>0, then a n>0 for any natural number 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 odd number 2 m−1 , then 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.

Let us immediately note that all written equalities are identical subject to the specified conditions, both their right and left parts can be swapped. For example, the main property of the fraction a m ·a n =a m+n with simplifying 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 power 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 a power of the number a with a natural exponent m+n, that is, a m+n. This completes the proof.

Let us give an example confirming the main property of the degree. Let's take degrees with the same bases 2 and natural powers 2 and 3, using the basic property of degrees we can write the equality 2 2 ·2 3 =2 2+3 =2 5. Let's check its validity by calculating the values ​​of the expressions 2 2 · 2 3 and 2 5 . Carrying out exponentiation, we have 2 2 ·2 3 =(2·2)·(2·2·2)=4·8=32 and 2 5 =2·2·2·2·2=32, since equal values ​​are obtained, then the equality 2 2 ·2 3 =2 5 is correct, and it confirms the main property of the degree.

The basic property of a degree, based on the properties of multiplication, can be generalized to the product of three or more powers with the same bases and natural exponents. So for any number k of natural numbers n 1, n 2, …, n k the following equality is true: 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 .

We can move on to the next property of powers with a natural exponent – property of quotient 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 presenting 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 we cannot divide by zero. The condition m>n is introduced so that we do not go beyond the 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 resulting equality a m−n ·a n =a m and it follows that a m−n is a quotient of the powers a m and a n . This proves the property of quotient powers with identical bases.

Let's give an example. Let's take two degrees with the same bases π and natural exponents 5 and 2, the equality π 5:π 2 =π 5−3 =π 3 corresponds to the considered property of the degree.

Now let's consider product power property: the natural power n of the product of any two real numbers a and b is equal to the product of the powers a n and b n , that is, (a·b) n =a n ·b n .

Indeed, by the 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 .

Here's an example: .

This property extends to the power of the product of three or more factors. That is, the property of natural degree 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 will show this property with an example. For the product of three factors to the power of 7 we have .

The following property is property of a quotient in kind: the quotient of real numbers a and b, b≠0 to the natural power n is equal to the quotient of 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 from the equality (a:b) n ·b n =a n it follows that (a:b) n is the quotient of a n divided by b n .

Let's write this property using specific numbers as an example: .

Now let's voice it property of raising a power to a power: 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 the number 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-to-degree property is the following chain of equalities: .

The property considered can be extended to degree to degree to degree, etc. 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.

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

First, let's prove 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 suggest that the result of multiplying any number of positive numbers will also be a positive number. And the power of a number a with natural exponent n, by definition, is 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 a n is a positive number. Due to the proven property 3 5 >0, (0.00201) 2 >0 and .

It is quite obvious that for any natural number 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 of degree.

Let's start with the case when the exponent is an even number, let's 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 moduli of the numbers a and a, which means it is a positive number. Therefore, the product will also be positive and degree a 2·m. Let's give examples: (−6) 4 >0 , (−2,2) 12 >0 and .

Finally, when the base 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 .

Let's move on to the property of comparing powers with the same natural exponents, which has the following formulation: of two powers with the same natural exponents, n is less than the one whose base is smaller, and greater is the one whose base is larger. Let's prove it.

Inequality a n properties of inequalities a provable inequality of the form a n is also true (2.2) 7 and .

It remains to prove the last of the listed properties of powers with natural exponents. Let's formulate it. Of two powers with natural exponents and identical positive bases less than one, the one whose exponent is smaller is greater; and of two powers with natural exponents and identical bases greater than one, the one whose exponent is greater is greater. Let us proceed to the proof of this property.

Let us prove that for m>n and 0 0 due to the initial condition m>n, which means 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 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 a m−n is greater than one . Consequently, a m −a n >0 and a m >a n , which is what needed to be proven. This property is illustrated by the inequality 3 7 >3 2.

Properties of powers with integer exponents

Since positive integers are natural numbers, then all the properties of powers with positive integer exponents coincide exactly with the properties of powers with natural exponents listed and proven in the previous paragraph.

We defined a degree with an integer negative exponent, as well as a degree with a zero exponent, in such a way that all properties of degrees with natural exponents, expressed by equalities, remained valid. Therefore, all these properties are valid for both zero exponents and negative exponents, while, of course, the bases of the powers are different from zero.

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 powers with integer exponents:

1. a m ·a n =a m+n ;
2. a m:a n =a m−n ;
3. (a·b) n =a n ·b n ;
4. (a:b) n =a n:b n ;
5. (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 holds.

When 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.

Proving each of these properties is not difficult; to do this, it is enough to use the definitions of degrees with natural and integer exponents, as well as the properties of operations with real numbers. As an example, let us prove that the power-to-power property holds for both positive integers and non-positive integers. To do this, you 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 proven in the previous paragraph. 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.

Now we prove that (a −p) q =a (−p)·q . By definition of a power with a negative integer exponent, then . By the property of quotients to powers we have . Since 1 p =1·1·…·1=1 and , then . The last expression, by definition, is a power of the form a −(p·q), which, due to the rules of multiplication, can be written as a (−p)·q.

Likewise .

AND .

Using the same principle, you 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 valid for any negative integer −n and any positive a and bfor which the condition a is satisfied . 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 the quotient of the positive numbers b n −a n and a n ·b n . Therefore, whence a −n >b −n , which is what needed to be proved.

The last property of powers with integer exponents is proved in the same way as a similar property of powers with natural exponents.

Properties of powers with rational exponents

We defined a degree with a fractional exponent by extending the properties of a degree with an integer exponent to it. In other words, powers with fractional exponents have the same properties as powers 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, and on the properties of a degree with an integer exponent. Let us provide evidence.

By definition of a power 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 , from which, by the definition of a degree with a fractional exponent, we have , and the indicator of the degree obtained can be transformed as follows: . This completes the proof.

The second property of powers with fractional exponents is proved in an absolutely similar way:

The remaining equalities are proved using similar principles:

Let's move on to proving the next property. Let us prove that for any positive a and b, a b p . Let's 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 the conditions m<0 и m>0 accordingly. For m>0 and a

Similarly, for m<0 имеем a m >b m , from where, 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 at 0 0 – inequality a p >a q . We can always reduce rational numbers p and q to a common denominator, even if we 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 the properties of the roots can be rewritten accordingly as And . And the definition of a degree with a rational exponent allows us to move on to inequalities and, accordingly. From here we draw the final conclusion: for p>q and 0 0 – inequality a p >a q .

Properties of powers with irrational exponents

From the way a degree with an irrational exponent is defined, we can conclude 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 powers 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 textbook for 5th grade. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 7th grade. educational institutions.
• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. 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 beginnings of analysis: Textbook for grades 10 - 11 of general education institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).

Power with rational exponent

Khasyanova T.G.,

mathematics teacher

The presented material will be useful to mathematics teachers when studying the topic “Exponent with a rational exponent.”

The purpose of the presented material: to reveal my experience of conducting a lesson on the topic “Exponent with a rational exponent” work program discipline "Mathematics".

The methodology for conducting the lesson corresponds to its type - a lesson in studying and initially consolidating new knowledge. Updated background knowledge and skills based on previously gained experience; primary memorization, consolidation and application of new information. Consolidation and application of new material took place in the form of solving problems that I tested of varying complexity, giving a positive result in mastering the topic.

At the beginning of the lesson, I set the following goals for the students: educational, developmental, educational. During the lesson I used various ways activities: frontal, individual, pair, independent, test. The tasks were differentiated and made it possible to identify, at each stage of the lesson, the degree of knowledge acquisition. The volume and complexity of tasks corresponds to the age characteristics of students. From my experience - homework, similar to the problems solved in the classroom, allows you to reliably consolidate the acquired knowledge and skills. At the end of the lesson, reflection was carried out and the work of individual students was assessed.

The goals were achieved. Students studied the concept and properties of a degree with a rational exponent, and learned to use these properties when solving practical problems. Behind independent work Grades will be announced at the next lesson.

I believe that the methodology I use for teaching mathematics can be used by mathematics teachers.

Lesson topic: Power with rational exponent

The purpose of the lesson:

Identifying the level of students’ mastery of a complex of knowledge and skills and, on its basis, applying certain solutions to improve the educational process.

Lesson objectives:

Educational: to form new knowledge among students of basic concepts, rules, laws for determining degrees with a rational indicator, the ability to independently apply knowledge in standard conditions, in modified and non-standard conditions;

developing: think logically and implement Creative skills;

raising: develop interest in mathematics, replenish vocabulary with new terms, gain Additional information about the world around us. Cultivate patience, perseverance, and the ability to overcome difficulties.

Organizing time

Updating of reference knowledge

When multiplying powers with the same bases, the exponents are added, but the base remains the same:

For example,

2. When dividing degrees with the same bases, the exponents of the degrees are subtracted, but the base remains the same:

For example,

3. When raising a degree to a power, the exponents are multiplied, but the base remains the same:

For example,

4. The degree of the product is equal to the product of the degrees of the factors:

For example,

5. The degree of the quotient is equal to the quotient of the degrees of the dividend and divisor:

For example,

Exercises with solutions

Find the meaning of the expression:

Solution:

IN in this case In explicit form, none of the properties of a degree with a natural exponent can be applied, since all degrees have different reasons. Let's write some powers in a different form:

(the degree of the product is equal to the product of the degrees of the factors);

(when multiplying powers with the same bases, the exponents are added, but the base remains the same; when raising a degree to a power, the exponents are multiplied, but the base remains the same).

Then we get:

In this example, the first four properties of a degree with a natural exponent were used.

Arithmetic square root
is a non-negative number whose square is equal toa,
. At
- expression
not defined, because there is no real number whose square is equal to a negative numbera.

Mathematical dictation(8-10 min.)

Option

II. Option

1.Find the value of the expression

A)

b)

1.Find the value of the expression

A)

b)

2.Calculate

A)

b)

IN)

2.Calculate

A)

b)

V)

Self-test(on the lapel board):

Response Matrix:

№ option/task

Problem 1

Problem 2

Option 1

a) 2

b) 2

a) 0.5

b)

V)

Option 2

a) 1.5

b)

A)

b)

at 4

II. Formation of new knowledge

Let's consider what meaning the expression has, where - positive number– fractional number and m-integer, n-natural (n›1)

Definition: power of a›0 with rational exponentr = , So, by definition we get that-whole, n-natural ( n›1) the number is called.

So:

For example:

Notes:

1. For any positive a and any rational r number positively.

2. When
rational degree numbersanot determined.

Expressions like
don't make sense.

3.If a fractional positive number is
.

If fractional negative number, then -doesn't make sense.

For example: - doesn't make sense.

Let's consider the properties of a degree with a rational exponent.

Let a >0, b>0; r, s - any rational numbers. Then a degree with any rational exponent has the following properties:

1.
2.
3.
4.
5.

III. Consolidation. Formation of new skills and abilities.

Task cards work in small groups in the form of a test.

From integer exponents of the number a, the transition to rational exponents suggests itself. Below we will define a degree with a rational exponent, and we will do this in such a way that all the properties of a degree with an integer exponent are preserved. This is necessary because integers are part of the rational numbers.

It is known that the set of rational numbers consists of integers and fractions, and each fraction can be represented as a positive or negative ordinary fraction. We defined a degree with an integer exponent in the previous paragraph, therefore, in order to complete the definition of a degree with a rational exponent, we need to give meaning to the degree of the number a with a fractional indicator m/n, Where , Where is an integer, and is an integer, and- natural. Let's do it.

Let's consider a degree with a fractional exponent of the form . For the power-to-power property to remain valid, the equality must hold . If we take into account the resulting equality and how we determined the nth root of the degree, then it is logical to accept, provided that given the given , Where, is an integer, and And a the expression makes sense.

It is easy to check that for all properties of a degree with an integer exponent are valid (this was done in the section properties of a degree with a rational exponent).

The above reasoning allows us to make the following conclusion: if given data , Where, is an integer, and And a the expression makes sense, then the power of the number a with a fractional indicator m/n called the root is an integer, and th degree of a to a degree , Where.

This statement brings us close to the definition of a degree with a fractional exponent. All that remains is to describe at what , Where, is an integer, and And a the expression makes sense. Depending on the restrictions imposed on , Where, is an integer, and And a There are two main approaches.

1. The easiest way is to impose a restriction on a, having accepted a≥0 for positive , Where And a>0 for negative , Where(since when m≤0 degree 0 m not determined). Then we get the following definition of a degree with a fractional exponent.

Definition.

Power of a positive number a with a fractional indicator m/n , Where , Where- whole, and is an integer, and– a natural number, called a root is an integer, and-th of the number a to a degree , Where, that is, .

The fractional power of zero is also determined with the only caveat that the indicator must be positive.

Definition.

Power of zero with fractional positive exponent m/n , Where , Where is a positive integer, and is an integer, and– natural number, defined as .
When the degree is not determined, that is, the degree of the number zero with a fractional negative exponent does not make sense.

It should be noted that with this definition of a degree with a fractional exponent, there is one caveat: for some negative a and some , Where And is an integer, and the expression makes sense, but we discarded these cases by introducing the condition a≥0. For example, the entries make sense or , and the definition given above forces us to say that powers with a fractional exponent of the form do not make sense, since the base should not be negative.

2. Another approach to determining the degree with a fractional exponent m/n consists in separately considering even and odd exponents of the root. This approach requires an additional condition: the power of the number a, the exponent of which is a reducible ordinary fraction, is considered a power of the number a, the indicator of which is the corresponding irreducible fraction (the importance of this condition will be explained below). That is, if m/n is an irreducible fraction, then for any natural number k degree is preliminarily replaced by .

For even is an integer, and and positive , Where the expression makes sense for any non-negative a(an even root of a negative number has no meaning), for negative , Where number a must still be different from zero (otherwise there will be division by zero). And for odd is an integer, and and positive , Where number a can be any (an odd root is defined for any real number), and for negative , Where number a must be non-zero (so that there is no division by zero).

The above reasoning leads us to this definition of a degree with a fractional exponent.

Definition.

Let m/n– irreducible fraction, , Where- whole, and is an integer, and- natural number. For any reducible fraction, the degree is replaced by . Degree of a with an irreducible fractional exponent m/n- it is for

o any real number a, whole positive , Where and odd natural is an integer, and, For example, ;

o any non-zero real number a, negative integer , Where and odd is an integer, and, For example, ;

o any non-negative number a, whole positive , Where and even is an integer, and, For example, ;

o any positive a, negative integer , Where and even is an integer, and, For example, ;

o in other cases, the degree with a fractional indicator is not determined, as for example the degrees are not defined .a we do not attach any meaning to the entry; we define the power of the number zero for positive fractional exponents m/n How , for negative fractional exponents the power of the number zero is not determined.

In conclusion of this paragraph, let us pay attention to the fact that the fractional exponent can be written in the form decimal or mixed number, For example, . To calculate the values ​​of expressions of this type, you need to write the exponent in the form of an ordinary fraction, and then use the definition of the exponent with a fractional exponent. For the above examples we have And