Degree and its properties. The Comprehensive Guide (2019). Power with rational exponent

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 : "Degree with rational indicator».

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 valid:

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 – natural ( n > 1), called the number m .

So, by definition we get that = m .

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 powers 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. In preparation homework ( No. 428 and No. 429

The video lesson “Exponent with a rational exponent” contains a visual educational material to teach a lesson 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 purpose of this video lesson is to clearly and clearly present the educational material, facilitate its development and memorization by students, and develop the ability to solve problems using the learned concepts.

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

The video lesson begins by introducing the topic. Linking studies new topic with previously studied material, it is suggested to remember that n √a is otherwise denoted by a 1/n for natural n and positive a. This n-root representation is displayed on the screen. Next, we propose to consider what the expression a m/n means, in which a is a positive number and m/n is a fraction. The definition of a degree with a rational exponent as a m/n = n √a m is given, highlighted in the frame. It is noted that n can be a natural number, and m can be an integer.

After defining a degree with a rational exponent, its meaning is revealed through examples: (5/100) 3/7 = 7 √(5/100) 3. An example is also shown in which a power represented by a decimal is converted to ordinary fraction to be represented as a root: (1/7) 1.7 =(1/7) 17/10 = 10 √(1/7) 17 and example with negative value degrees: 3 -1/8 = 8 √3 -1.

The peculiarity of the special case when the base of the degree is zero is indicated separately. It is noted that this degree makes sense only with a positive fractional exponent. In this case, its value is zero: 0 m/n =0.

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

Next in the video lesson we discuss the properties of a degree with a rational exponent. 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 on 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 the exponents: a p:a q =a p-q.
3. If we raise the degree to a certain power, then we end up with a degree with a given base and the product of 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 when opening parentheses remain true:

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

The video tutorial discusses solving examples that use the considered properties of powers with a rational exponent. The first example asks you to find the value of an expression that contains variables x in 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 powers it can be solved quite simply. Solving the problem begins with simplifying 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 obtain the value - 50.

In the second example, you need to reduce a fraction whose numerator and denominator contain powers with a rational exponent. Using the properties of the degree, we extract from the difference the factor x 1/3, which is then reduced in the numerator and denominator, and using the formula for the difference of squares, the numerator is factorized, which gives further reductions of identical factors in the numerator and denominator. The result of such transformations is the short fraction x 1/4 +3.

The video lesson “Exponent with a rational exponent” can be used instead of the teacher explaining a new lesson topic. Also this manual contains enough full information For self-study student. The material can also be useful for distance learning.

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. Basic knowledge and skills were updated on the basis of previously gained experience; primary memorization, consolidation and application of new information. The consolidation and application of new material took place in the form of solving problems that I tested of varying complexity, giving positive result 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 age characteristics 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 realize creative abilities;

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 this case, none of the properties of a degree with a natural exponent can be applied explicitly, 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 in this example The first four properties of degree with natural exponent were used.

Arithmetic square root
is a non-negative number whose square is equal toa,
. At
- expression
not defined, because there's no such thing 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 – a fractional number and m-integer, n-natural (n›1)

Definition: power of a›0 with rational exponentr = , m-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 power of a numberanot 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.

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. The universal way to read the notation 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 the power by known value degree and known indicator. This task leads to .

It is known that the set of rational numbers consists of integers and fractions, and each 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.

Degree 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 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 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 .

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 m is an integer, and n- 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 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 data m, n And a the expression makes sense, then the power of the number a with a fractional indicator m/n called the root n th degree of a to a degree 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 imposed on m, n And a There are two main approaches.

1. The easiest way is to impose a restriction on a, having accepted a≥0 for positive m And a>0 for negative m(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 m- whole, and n– a natural number, called a root n-th of the number a to a degree 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– 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 m And n 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 n and positive m the expression makes sense for any non-negative a(an even root of a negative number has no meaning), for negative m number a must still be different from zero (otherwise there will be division by zero). And for odd n and positive m number a can be any (an odd root is defined for any real number), and for negative m 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, m- whole, and n- 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 m and odd natural n, For example, ;

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

o any non-negative number a, whole positive m and even n, For example, ;

o any positive a, negative integer m and even n, 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 a 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