Degrees with rational and irrational exponents are examples. Degree of number: definitions, designation, examples

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

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Degree 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 degree of a with natural exponent n is given for a , which we will call base of degree, and n , which we will call exponent. Also note that the degree with a natural indicator is determined through the product, so to understand the material below, you need to have an idea about the multiplication of numbers.

Definition.

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

Immediately it is worth mentioning the rules for reading degrees. Universal way reading the entry a n is: "a to the power of n". In some cases, such options are also acceptable: "a to the nth power" and "nth power of the number a". For example, let's take the power of 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 the square of a number, for example, 7 2 is read as "seven squared" or "square of the number seven". The third power of a number is called cube number, for example, 5 3 can be read as "five cubed" or say "cube of the number 5".

It's time to bring examples of degrees with physical indicators. Let's start with the power of 5 7 , where 5 is the base of the power and 7 is the exponent. Let's give another example: 4.32 is the base, and the natural number 9 is the exponent (4.32) 9 .

Please note that in the last example, the base of the degree 4.32 is written in brackets: to avoid discrepancies, we will take in brackets all the bases of the degree that are different from natural numbers. As an example, we give the following degrees with natural indicators , 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 the records of the form (−2) 3 and −2 3 . The expression (−2) 3 is the power of −2 with natural exponent 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 degree of a with an exponent n of the form a^n . Moreover, if n is a multivalued 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 more examples of writing degrees using the “^” symbol: 14^(21) , (−2,1)^(155) . In what follows, we will mainly use the notation of the degree of the form a n .

One of the problems inverse to exponentiation with a natural exponent is the problem of finding the base of the degree by known value degree and known exponent. This task leads to .

It is known that the set of rational numbers consists of integers and fractional numbers, and each fractional number can be represented as positive or negative common fraction. We defined the degree with an integer exponent in the previous paragraph, therefore, in order to complete the definition of the degree with rational indicator, you need to give the meaning of 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.

Consider a degree with a fractional exponent of the form . In order for the property of degree in a degree to remain valid, the equality must hold . If we take into account the resulting equality and the way we defined , then it is logical to accept, provided that for given m, n and a, the expression makes sense.

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

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

This statement brings us close to the definition of a degree with a fractional exponent. It remains only to describe for which m, n and a the expression makes sense. Depending on the restrictions imposed on m , n and a, there are two main approaches.

The easiest way to constrain a is to assume a≥0 for positive m and a>0 for negative m (because m≤0 has no power of 0 m). Then we get the following definition of the 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 root of the nth of the number a to the power of m, that is, .

The fractional degree of zero is also defined with the only caveat that the exponent 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 defined, that is, the degree of the number zero with a fractional negative exponent does not make sense.

It should be noted that with such a definition of the degree with a fractional exponent, there is one nuance: 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, it makes sense to write or , and the above definition forces us to say that degrees with a fractional exponent of the form are meaningless, since the base must not be negative.

Another approach to determining the degree with a fractional exponent m / n is to separately consider the even and odd exponents of the root. This approach requires additional condition: the degree of the number a, the exponent of which is, is considered the degree of the number a, the exponent 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 the degree is first replaced by .

For even n and positive m, the expression makes sense for any non-negative a (the root of an even degree from a negative number does not make sense), for negative m, the number a must still be different from zero (otherwise there will be a division by zero). And for odd n and positive m, the number a can be anything (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 such a definition of the degree with a fractional exponent.

Definition.

Let m/n be an irreducible fraction, m an integer, and n a natural number. For any reducible ordinary fraction, the degree is replaced by . The power of a 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 encounter situations similar to the following: since 6/10=3/5 , then the equality , but , a .

The video lesson "Degree with a rational indicator" contains a visual educational material to teach on this topic. The video lesson contains information about the concept of a degree with a rational exponent, properties of such degrees, as well as examples describing the use of educational material to solve practical problems. The task of this video lesson is to clearly and clearly present the educational material, to facilitate its development and memorization by students, to form the ability to solve problems using the concepts learned.

The main advantages of the video lesson are the ability to make visual transformations and calculations, the ability to use animation effects to improve learning efficiency. Voice accompaniment helps to develop correct mathematical speech, and also makes it possible to replace the teacher's explanation, freeing him for individual work.

The video tutorial starts by introducing the topic. Linking study new topic with the previously studied material, it is suggested to recall that n √ a is otherwise denoted by a 1/n for natural n and positive a. This representation of the n-root is displayed on the screen. Further, it is proposed to consider what the expression a m / n means, in which a is a positive number, and m / n is some fraction. The definition of the degree highlighted in the box is given with a rational exponent as a m/n = n √ a m . It is noted that n can be a natural number, and m - an integer.

After determining the degree with a rational exponent, its meaning is revealed by examples: (5/100) 3/7 = 7 √(5/100) 3 . An example is also shown where a power represented by a decimal is converted to a common fraction to be represented as a root: (1/7) 1.7 =(1/7) 17/10 = 10 √(1/7) 17 and an example from negative value degrees: 3 -1/8 \u003d 8 √3 -1.

Separately, a feature of a particular case is indicated when the base of the degree is zero. It is noted that this degree makes sense only with a positive fractional exponent. In this case, its value is equal to zero: 0 m/n =0.

Another feature of the degree with a rational exponent is noted - that the degree with a fractional exponent cannot be considered with a fractional exponent. Examples of incorrect notation of the degree are given: (-9) -3/7 , (-3) -1/3 , 0 -1/5 .

Further in the video lesson, the properties of a degree with a rational exponent are considered. It is noted that the properties of a degree with an integer exponent will also be valid for a degree with a rational exponent. It is proposed to recall the list of properties that are also valid in this case:

1. When multiplying powers with the same grounds their indicators add up: a p a q =a p+q .
2. The division of degrees with the same bases is reduced to a degree with a given base and the difference in exponents: a p:a q =a p-q .
3. If we raise the power to a certain power, then as a result we get the power with the given base and the product of the exponents: (a p) q =a pq .

All these properties are valid for powers with rational exponents p, q and positive base a>0. Also, degree transformations remain true when opening parentheses:

1. (ab) p =a p b p - raising a product of two numbers to a certain power with a rational exponent is reduced to a product of numbers, each of which is raised to a given power.
2. (a/b) p =a p /b p - exponentiation with a rational exponent of a fraction is reduced to a fraction whose numerator and denominator are raised to the given power.

The video tutorial discusses the solution of examples that use the considered properties of degrees with a rational exponent. In the first example, it is proposed to find the value of an expression that contains the variables x to a fractional power: (x 1/6 -8) 2 -16x 1/6 (x -1/6 -1). Despite the complexity of the expression, using the properties of degrees, it is solved quite simply. The solution of the task begins with a simplification of the expression, which uses the rule of raising a power with a rational exponent to a power, as well as multiplying powers with the same base. After substituting the given value x=8 into the simplified expression x 1/3 +48, ​​it is easy to get the value - 50.

In the second example, it is required to reduce a fraction whose numerator and denominator contain powers with a rational exponent. Using the properties of the degree, we select the factor x 1/3 from the difference, which is then reduced in the numerator and denominator, and using the difference of squares formula, the numerator is decomposed into factors, which gives more reductions of the same factors in the numerator and denominator. The result of such transformations is a short fraction x 1/4 +3.

The video lesson "Degree with a rational indicator" can be used instead of the teacher explaining the new topic of the lesson. Also, this manual contains sufficient information for self-study student. The material can be useful in distance learning.

MBOU "Sidorskaya

comprehensive school»

Development of a plan-outline open lesson

in algebra in grade 11 on the topic:

Prepared and conducted

math teacher

Iskhakova E.F.

Outline of an open lesson in algebra in grade 11.

Subject : "Degree with a rational exponent".

Lesson type : Learning new material

Lesson Objectives:

To acquaint students with the concept of a degree with a rational indicator and its main properties, based on previously studied material (a degree with an integer indicator).

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

To cultivate mathematical literacy and mathematical interest in students.

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

During the classes:

Organizing time.

Checking the assimilation of the topic covered by individual task cards.

Task number 1.

=2;

B) = x + 5;

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

4 - 5 =6.

Task number 2.

Solve the irrational equation: = - 3;

B) = x - 2;

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

3 - 2 = - 2.

Presentation of the topic and objectives of the lesson.

The topic of our today's lesson Degree with rational exponent».

Explanation of new material on the example of previously studied.

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

Repetition with Presentation Degree with integer exponent».

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

a m * a n = a m + n ;

a m: a n = a m-n (a ≠ 0);

(am) 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 the degree of a number and give meaning to expressions that have a fractional exponent. Let's introduce definition degrees with a rational indicator (Presentation "Degree with a rational indicator"):

The degree of a > 0 with a rational exponent r = , where m is an integer, and n - natural ( n > 1), called the number m .

So, by definition, we get that = m .

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

EXAMPLE #1

I Express as a root of a number the expression:

BUT) B) AT) .

Now let's try to apply this definition in reverse

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

BUT) 2 B) AT) 5 .

The power of 0 is only defined 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 the above definition of a degree with a rational exponent preserves the basic properties of degrees that are true for any exponent.

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 on the use of several properties at once: * : .

Fizkultminutka.

We put pens on the desk, straightened the backs, and now we are reaching forward, we want to touch the board. And now we lifted and leaned to the right, to the left, forward, back. They showed me the pens, and now show me how your fingers can dance.

Work on the material

We note two more properties of powers with rational exponents:

60 . Let be 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 numbers:

And ; 2 300 and 3 200 .

Lesson summary:

Today in the lesson we remembered the properties of a degree with an integer exponent, learned the definition and basic properties of a degree with a rational exponent, considered the application of this theoretical material in practice during exercise. I want to draw your attention to the fact that the topic "Degree with a rational indicator" is mandatory in USE assignments. When preparing homework No. 428 and No. 429

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 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 integers, that m>n , then for 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.

The degree with a negative integer exponent, as well as the degree with a zero exponent, we defined in such a way that all properties of degrees with natural exponents expressed by equalities remain valid. Therefore, all these properties are valid both for zero exponents and for negative exponents, while, of course, the bases of the degrees are nonzero.

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

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

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 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 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 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).

Degree with rational exponent

Khasyanova T.G.,

mathematics teacher

The presented material will be useful to teachers of mathematics when studying the topic "Degree with a rational indicator".

The purpose of the presented material: disclosure of my experience in conducting a lesson on the topic "Degree with a rational indicator" work program discipline "Mathematics".

The methodology of the lesson corresponds to its type - a lesson in the study and primary consolidation of new knowledge. The basic knowledge and skills were updated on the basis of 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 of varying complexity that I tested, giving positive result mastering the topic.

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

The goals have been achieved. The students studied the concept and properties of a degree with a rational exponent, learned how to use these properties in solving practical problems. Behind independent work grades are announced in the next lesson.

I believe that the methodology used by me for conducting classes in mathematics can be applied by teachers of mathematics.

Lesson topic: Degree with a rational indicator

The purpose of the lesson:

Identification of the level of mastering by students of a complex of knowledge and skills and, on its basis, the application of certain solutions to improve the educational process.

Lesson objectives:

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

developing: think logically and implement Creative skills;

educators: to form an interest in mathematics, replenish vocabulary with new terms, get Additional information about the world around. Cultivate patience, perseverance, the ability to overcome difficulties.

Organizing time

Updating of basic knowledge

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

For example,

2. When dividing powers with the same bases, the exponents are subtracted, and the base remains the same:

For example,

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

For example,

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

For example,

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

For example,

Solution Exercises

Find the value of an expression:

Decision:

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

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

(when multiplying powers with the same base, the exponents are added, and 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:

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

Arithmetic square root
is a non-negative number whose square isa,
. 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)

AT)

2. Calculate

a)

b)

in)

Self test(on the lapel board):

Response Matrix:

№ option/task

Task 1

Task 2

Option 1

a) 2

b) 2

a) 0.5

b)

in)

Option 2

a) 1.5

b)

a)

b)

at 4

II. Formation of new knowledge

Consider the meaning of the expression, where - positive number– fractional number and m-integer, n-natural (n>1)

Definition: degree of number a›0 with rational exponentr = , m-whole, n- natural ( n›1) a number is called.

So:

For example:

Notes:

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

2. When
rational power of a numberanot defined.

Expressions such as
don't make sense.

3.If fractional positive number
.

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

For example: - doesn't make sense.

Consider the properties of a degree with a rational exponent.

Let a>0, в>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.

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