From integer exponents of the number a, the transition to rational indicator. Below we define a degree with a rational exponent, and we will do it 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 rational numbers.

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 a rational exponent, we need to give a meaning to the degree of the number a With fractional indicator m/n, where m is an integer, and n- natural. 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 how we determined the root of the nth degree, then it is logical to accept, provided that with the data m, n and a the expression makes sense.

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

The above reasoning allows us to make the following conclusion: if given m, n and a expression makes sense, then the power of the number a with a fraction m/n called the root n th degree of a to the extent m.

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

1. The easiest way is to impose a restriction on a, accepting a≥0 for positive m and a>0 for negative m(because at m≤0 degree 0 m not determined). Then we get the following definition of the degree with a fractional exponent.

Definition.

Degree of a positive number a with a fraction m/n , where m is a whole, and n is a natural number, called a root n-th from among a to the extent 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, defined as .
When the degree is not defined, that is, the degree of zero with a fractional negative indicator doesn't 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.

2. Another approach to determining the degree with a fractional exponent m/n consists in separate consideration of even and odd exponents of the root. This approach requires additional condition: degree of a, whose indicator is a reduced ordinary fraction, is considered a power of a number a, whose indicator 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 n and positive m expression makes sense for any non-negative a(root even degree from a negative number does not make sense), with negative m number a must still be different from zero (otherwise it will be a division by zero). And for odd n and positive m number a can be anything (an odd root is defined for any real number), and for negative m 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- irreducible fraction m is a whole, and n- natural number. For any reducible ordinary fraction, the degree is replaced by . Degree of a with irreducible fractional exponent m/n- it's for

o any real number a, an integer positive m and odd natural n, for example, ;

o any non-zero real number a, an integer negative m and odd n, for example, ;

o any non-negative number a, an integer positive m and even n, for example, ;

o any positive a, an integer negative m and even n, for example, ;

o in other cases, the degree with a fractional exponent is not defined, as, for example, degrees are not defined .a entries we do not attach any meaning, we define the degree of zero for positive fractional exponents m/n how , for negative fractional exponents, the degree of the number zero is not defined.

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

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.

Page navigation.

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 indicator 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. The universal way to read 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 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 a positive or negative ordinary fraction. We defined the degree with an integer exponent in the previous paragraph, therefore, in order to complete the definition of the degree with a rational exponent, we 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 check that all properties of a degree with an integer exponent are valid for as (this is done in the section on the 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 an additional condition: the degree of the number a, whose exponent 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 non-zero (otherwise division by zero will occur). 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 .

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. An update has been made basic knowledge and skills based on previous experience; primary memorization, consolidation and application of new information. Consolidation and application of new material took place in the form of solving problems I tested of varying complexity giving a positive result of 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 is 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. Students studied the concept and properties of a degree with a rational exponent, learned how to use these properties when solving practical tasks. Per 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, application certain decisions 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: develop interest in mathematics vocabulary 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:

Solution:

AT this case in explicit form, none of the properties of a degree with a natural exponent can be applied, 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 not a 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 degree numbersanot 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.

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.

Topic : "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, at home you will complete #428 and #429.

Let us now show that the above definition of a degree with a rational exponent retains 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: *

twenty . 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 r- 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. In preparation homework ( No. 428 and No. 429

The video lesson "Degree with a rational indicator" contains a visual educational material to teach on this topic. The video tutorial 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 visually 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 guidance helps develop the 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 the root of the n-power 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 natural number, and m is 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 in which the degree represented by decimal, is converted to fraction to be represented as a root: (1/7) 1.7 =(1/7) 17/10 = 10 √(1/7) 17 and example c 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 bases, their indicators are added up: a p a q \u003d 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. The first example proposes to find the value of the expression that contains the variables x in fractional degree: (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 degree with a rational exponent to a power, as well as multiplying degrees with the same base. After substitution set value x=8 into a simplified expression x 1/3 +48, ​​it's 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 enough full information for self-study student. The material can be useful in distance learning.