Lesson type: lesson of generalization and systematization of knowledge

Goals:

• educational– repeat the definition of a degree, the rules for multiplying and dividing degrees, raising a degree to a power, consolidate the skills of solving examples containing degrees,
• developing- development logical thinking students, interest in the material being studied,
• raising– fostering a responsible attitude to learning, a culture of communication, and a sense of collectivism.

Equipment: computer, multimedia projector, interactive whiteboard, presentation of “Degrees” for mental calculation, cards with tasks, handouts.

Lesson plan:

1. Organizational moment.
2. Repetition of rules
3. Oral counting.
4. Historical information.
5. Work at the board.
6. Physical education minute.
7. Working on an interactive whiteboard.
8. Independent work.
9. Homework.
10. Summing up the lesson.

Lesson progress

I. Organizational moment

Communicate the topic and objectives of the lesson.

In previous lessons you discovered amazing world degrees, learned to multiply and divide degrees, and raise them to a power. Today we must consolidate the acquired knowledge by solving examples.

II. Repetition of rules(orally)

1. Give the definition of degree with a natural exponent? (Power of number A with a natural exponent greater than 1 is called a product n factors, each of which is equal A.)
2. How to multiply two powers? (To multiply powers with the same bases, you must leave the base the same and add the exponents.)
3. How to divide degree by degree? (To divide powers with the same bases, you need to leave the base the same and subtract the exponents.)
4. How to raise a product to a power? (To raise a product to a power, you need to raise each factor to that power)
5. How to raise a degree to a power? (To raise a power to a power, you need to leave the base the same and multiply the exponents)

III. Oral counting(by multimedia)

IV. Historical background

All problems are from the Ahmes papyrus, which was written around 1650 BC. e. associated with construction practice, delimitation land plots etc. Tasks are grouped by topic. These are mainly tasks on finding the areas of a triangle, quadrilaterals and a circle, various operations with integers and fractions, proportional division, finding ratios, there is also raising in different degrees, solving equations of the first and second degree with one unknown.

There is a complete lack of any explanation or evidence. The desired result is either given directly or given short algorithm its calculations. This method of presentation, typical for science in countries ancient East, suggests that mathematics there developed through generalizations and guesses that did not form any general theory. However, the papyrus contains a number of evidence that Egyptian mathematicians knew how to extract roots and raise to powers, solve equations, and even mastered the rudiments of algebra.

V. Work at the board

Find the meaning of the expression in a rational way:

Calculate the value of the expression:

VI. Physical education minute

1. for the eyes
2. for the neck
3. for hands
4. for the torso
5. for feet

VII. Problem solving(with display on the interactive whiteboard)

Is the root of the equation a positive number?

a) 3x + (-0.1) 7 = (-0.496) 4 (x > 0)

b) (10.381) 5 = (-0.012) 3 - 2x (x< 0)

VIII. Independent work

IX. Homework

X. Summing up the lesson

Analysis of results, announcement of grades.

We will use the acquired knowledge about degrees when solving equations and problems in high school; they are also often found in the Unified State Exam.

Degree formulas used in the process of reducing and simplifying complex expressions, in solving equations and inequalities.

Number c is n-th power of a number a When:

Operations with degrees.

1. Multiplying powers of c the same basis their indicators add up:

a m·a n = a m + n .

2. When dividing degrees with the same base, their exponents are subtracted:

3. Power of the product of 2 or more factors is equal to the product of the powers of these factors:

(abc…) n = a n · b n · c n …

4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:

(a/b) n = a n /b n .

5. Raising a power to a power, the exponents are multiplied:

(a m) n = a m n .

Each formula above is true in the directions from left to right and vice versa.

For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.

Operations with roots.

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of a ratio is equal to the ratio of the dividend and the divisor of the roots:

3. When raising a root to a power, it is enough to raise the radical number to this power:

4. If you increase the degree of the root in n once and at the same time build into n th power is a radical number, then the value of the root will not change:

5. If you reduce the degree of the root in n extract the root at the same time n-th power of a radical number, then the value of the root will not change:

A degree with a negative exponent. The power of a certain number with a non-positive (integer) exponent is defined as one divided by the power of the same number with an exponent equal to absolute value non-positive indicator:

Formula a m:a n =a m - n can be used not only for m> n, but also with m< n.

For example. a4:a 7 = a 4 - 7 = a -3.

To formula a m:a n =a m - n became fair when m=n, the presence of zero degree is required.

A degree with a zero index. The power of any number not equal to zero with a zero exponent is equal to one.

For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.

Degree with a fractional exponent. To raise a real number A to the degree m/n, you need to extract the root n th degree of m-th power of this number A.

Sections: Mathematics

Lesson type: lesson of generalization and systematization of knowledge

Goals:

educational– repeat the definition of a degree, the rules for multiplying and dividing degrees, raising a degree to a power, consolidate the skills of solving examples containing degrees,

developing– development of students’ logical thinking, interest in the material being studied,

raising– fostering a responsible attitude to learning, a culture of communication, and a sense of collectivism.

Equipment: computer, multimedia projector, interactive whiteboard, presentation of “Degrees” for mental calculation, task cards, handouts.

Lesson plan:

Organizational moment.

Repetition of rules

Oral counting.

Historical information.

Work at the board.

Physical education minute.

Working on an interactive whiteboard.

Independent work.

Homework.

Summing up the lesson.

Lesson progress

I. Organizational moment

Communicate the topic and objectives of the lesson.

In previous lessons, you discovered the wonderful world of powers, learned how to multiply and divide powers, and raise them to powers. Today we must consolidate the acquired knowledge by solving examples.

II. Repetition of rules(orally)

1. Give the definition of degree with a natural exponent? (Power of number A with a natural exponent greater than 1 is called a product n factors, each of which is equal A.)
2. How to multiply two powers? (To multiply powers with the same bases, you must leave the base the same and add the exponents.)
3. How to divide degree by degree? (To divide powers with the same bases, you need to leave the base the same and subtract the exponents.)
4. How to raise a product to a power? (To raise a product to a power, you need to raise each factor to that power)
5. How to raise a degree to a power? (To raise a power to a power, you need to leave the base the same and multiply the exponents)

III. Oral counting(by multimedia)

IV. Historical background

All problems are from the Ahmes papyrus, which was written around 1650 BC. e. related to construction practice, demarcation of land plots, etc. Tasks are grouped by topic. These are mainly problems of finding the areas of a triangle, quadrilaterals and a circle, various operations with integers and fractions, proportional division, finding ratios, there is also raising to different powers, solving equations of the first and second degree with one unknown.

There is a complete lack of any explanation or evidence. The desired result is either given directly or a short algorithm for calculating it is given. This method of presentation, typical of science in the countries of the ancient East, suggests that mathematics there developed through generalizations and guesses that did not form any general theory. However, the papyrus contains a number of evidence that Egyptian mathematicians knew how to extract roots and raise to powers, solve equations, and even mastered the rudiments of algebra.

V. Work at the board

Find the meaning of the expression in a rational way:

Calculate the value of the expression:



VI. Physical education minute

6. for the eyes
7. for the neck
8. for hands
9. for the torso
10. for feet

VII. Problem solving(with display on the interactive whiteboard)

Is the root of the equation a positive number?

xn--i1abbnckbmcl9fb.xn--p1ai

Formulas of powers and roots.

Degree formulas used in the process of reducing and simplifying complex expressions, in solving equations and inequalities.

Number c is n-th power of a number a When:



Operations with degrees.

1. By multiplying degrees with the same base, their indicators are added:

2. When dividing degrees with the same base, their exponents are subtracted:



3. The degree of the product of 2 or more factors is equal to the product of the degrees of these factors:

(abc…) n = a n · b n · c n …

4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:

5. Raising a power to a power, the exponents are multiplied:

Each formula above is true in the directions from left to right and vice versa.

Operations with roots.

1. The root of the product of several factors is equal to the product of the roots of these factors:

2. The root of a ratio is equal to the ratio of the dividend and the divisor of the roots:



3. When raising a root to a power, it is enough to raise the radical number to this power:



4. If you increase the degree of the root in n once and at the same time build into n th power is a radical number, then the value of the root will not change:



5. If you reduce the degree of the root in n extract the root at the same time n-th power of a radical number, then the value of the root will not change:



The power of a certain number with a non-positive (integer) exponent is defined as one divided by the power of the same number with an exponent equal to the absolute value of the non-positive exponent:



Formula a m :a n =a m - n can be used not only for m > n, but also with m 4:a 7 = a 4 - 7 = a -3 .

To formula a m :a n =a m - n became fair when m=n, the presence of zero degree is required.

The power of any number not equal to zero with a zero exponent is equal to one.

To raise a real number A to the degree m/n, you need to extract the root n-th degree from m-th power of this number A:



Degree formulas.

6. a — n = - division of degrees;

7. - division of degrees;

8. a 1/n = ;

Degrees of the rule of action with degrees

1. The degree of the product of two or more factors is equal to the product of the degrees of these factors (with the same exponent):

(abc…) n = a n b n c n …

Example 1. (7 2 10) 2 = 7 2 2 2 10 2 = 49 4 100 = 19600. Example 2. (x 2 –a 2) 3 = [(x +a)(x – a)] 3 =( x +a) 3 (x - a) 3

In practice, the reverse conversion is more important:

a n b n c n … = (abc…) n

those. the product of identical powers of several quantities is equal to the same power of the product of these quantities.

Example 3. Example 4. (a +b) 2 (a 2 – ab +b 2) 2 =[(a +b)(a 2 – ab +b 2)] 2 =(a 3 +b 3) 2

2. The power of a quotient (fraction) is equal to the quotient of dividing the same power of the divisor by the same power:

Example 5. Example 6.

Reverse conversion:. Example 7. . Example 8. .

3. When multiplying degrees with the same bases, the exponents of the degrees are added:

Example 9.2 2 2 5 =2 2+5 =2 7 =128. Example 10. (a – 4c +x) 2 (a – 4c +x) 3 =(a – 4c + x) 5.

4. When dividing powers with the same bases, the exponent of the divisor is subtracted from the exponent of the dividend

Example 11. 12 5:12 3 =12 5-3 =12 2 =144. Example 12. (x-y) 3:(x-y) 2 =x-y.

5. When raising a degree to a power, the exponents are multiplied:

Example 13. (2 3) 2 =2 6 =64. Example 14.

www.maths.yfa1.ru

Powers and roots

Operations with powers and roots. Degree with negative ,

zero and fractional indicator. About expressions that have no meaning.

Operations with degrees.

1. When multiplying powers with the same base, their exponents are added:

a m · a n = a m + n .

2. When dividing degrees with the same base, their exponents are deducted .



3. The degree of the product of two or more factors is equal to the product of the degrees of these factors.

4. The degree of a ratio (fraction) is equal to the ratio of the degrees of the dividend (numerator) and divisor (denominator):

(a/b) n = a n / b n .

5. When raising a power to a power, their exponents are multiplied:

All the above formulas are read and executed in both directions from left to right and vice versa.

EXAMPLE (2 3 5 / 15)² = 2² · 3² · 5² / 15² = 900 / 225 = 4 .

Operations with roots. In all the formulas below, the symbol means arithmetic root(the radical expression is positive).

1. Root of the product of several factors equal to the product roots of these factors:

2. The root of a ratio is equal to the ratio of the roots of the dividend and the divisor:



3. When raising a root to a power, it is enough to raise to this power radical number:



4. If you increase the degree of the root by m times and at the same time raise the radical number to the mth power, then the value of the root will not change:



5. If you reduce the degree of the root by m times and simultaneously extract the mth root of the radical number, then the value of the root will not change:




Expanding the concept of degree. So far we have considered degrees only with natural exponents; but operations with powers and roots can also lead to negative, zero And fractional indicators. All these exponents require additional definition.

A degree with a negative exponent. The power of a certain number with a negative (integer) exponent is defined as one divided by the power of the same number with an exponent equal to the absolute value of the negative exponent:



Now the formula a m : a n = a m - n can be used not only for m, more than n, but also with m, less than n .

EXAMPLE a 4: a 7 =a 4 — 7 =a — 3 .

If we want the formula a m : a n = a m — n was fair when m = n, we need a definition of degree zero.

A degree with a zero index. The power of any non-zero number with exponent zero is 1.

EXAMPLES. 2 0 = 1, ( – 5) 0 = 1, (– 3 / 5) 0 = 1.

Degree with a fractional exponent. In order to raise a real number a to the power m / n, you need to extract the nth root of the mth power of this number a:



About expressions that have no meaning. There are several such expressions.

Where a ≠ 0 , does not exist.

In fact, if we assume that x is a certain number, then in accordance with the definition of the division operation we have: a = 0· x, i.e. a= 0, which contradicts the condition: a ≠ 0

— any number.

In fact, if we assume that this expression is equal to some number x, then according to the definition of the division operation we have: 0 = 0 · x. But this equality occurs when any number x, which was what needed to be proven.

0 0 — any number.



Solution. Let's consider three main cases:

1) x = 0 – this value does not satisfy this equation

2) when x> 0 we get: x/x= 1, i.e. 1 = 1, which means

What x– any number; but taking into account that in

in our case x> 0, the answer is x > 0 ;

Properties of degree

We remind you that in this lesson are sorting it out properties of degrees with natural indicators and zero. Powers with rational exponents and their properties will be discussed in lessons for 8th grade.

A degree with a natural indicator has several important properties, which allow you to simplify calculations in examples with powers.

Property No. 1
Product of powers

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

a m · a n = a m + n, where “a” is any number, and “m”, “n” are any natural numbers.

This property of powers also applies to the product of three or more powers.

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

Please note that in the specified property we were talking only about the multiplication of powers with the same bases. It does not apply to their addition.

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

Property No. 2
Partial degrees

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

• Write the quotient as a power
  (2b) 5: (2b) 3 = (2b) 5 − 3 = (2b) 2
• Calculate.

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

Answer: t = 3 4 = 81

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

Example. Simplify the expression.
4 5m + 6 4 m + 2: 4 4m + 3 = 4 5m + 6 + m + 2: 4 4m + 3 = 4 6m + 8 − 4m − 3 = 4 2m + 5

Example. Find the value of an expression using the properties of exponents.

2 11 − 5 = 2 6 = 64

Please note that in Property 2 we were only talking about dividing powers with the same bases.

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

Property No. 3
Raising a degree to a power

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

(a n) m = a n · m, where “a” is any number, and “m”, “n” are any natural numbers.

Example.
(a 4) 6 = a 4 6 = a 24

Example. Express 3 20 as a power with a base of 3 2.

By the property of raising a degree to a power It is known that when raised to a power, exponents are multiplied, which means:

Properties 4
Product power

When a power is raised to a product power, each factor is raised to that power and the results are multiplied.

(a b) n = a n b n, where “a”, “b” are any rational numbers; "n" - any natural number.

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

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

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

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

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

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

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

An example of raising a decimal to a power.

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

Properties 5
Power of a quotient (fraction)

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

(a: b) n = a n: b n, where “a”, “b” are any rational numbers, b ≠ 0, n - any natural number.

• Example. Present the expression as a quotient of powers.
  (5: 3) 12 = 5 12: 3 12

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

Entry level

Degree and its properties. Comprehensive guide (2019)

Why are degrees needed? Where will you need them? Why should you take the time to study them?

To learn everything about degrees, what they are for, how to use your knowledge in everyday life read this article.

And, of course, knowledge of degrees will bring you closer to successfully passing the Unified State Exam or Unified State Exam and to entering the university of your dreams.

Let's go... (Let's go!)

Important note! If you see gobbledygook instead of formulas, clear your cache. To do this, press CTRL+F5 (on Windows) or Cmd+R (on Mac).

ENTRY LEVEL

Exponentiation is a mathematical operation just like addition, subtraction, multiplication or division.

Now I will explain everything in human language using very simple examples. Be careful. The examples are elementary, but explain important things.

Let's start with addition.

There is nothing to explain here. You already know everything: there are eight of us. Everyone has two bottles of cola. How much cola is there? That's right - 16 bottles.

Now multiplication.

The same example with cola can be written differently: . Mathematicians are cunning and lazy people. They first notice some patterns, and then figure out a way to “count” them faster. In our case, they noticed that each of the eight people had the same number of cola bottles and came up with a technique called multiplication. Agree, it is considered easier and faster than.

So, to count faster, easier and without errors, you just need to remember multiplication table. Of course, you can do everything slower, more difficult and with mistakes! But…

Here is the multiplication table. Repeat.

And another, more beautiful one:

What other clever counting tricks have lazy mathematicians come up with? Right - raising a number to a power.

Raising a number to a power

If you need to multiply a number by itself five times, then mathematicians say that you need to raise that number to the fifth power. For example, . Mathematicians remember that two to the fifth power is... And they solve such problems in their heads - faster, easier and without mistakes.

All you need to do is remember what is highlighted in color in the table of powers of numbers. Believe me, this will make your life a lot easier.

By the way, why is it called the second degree? square numbers, and the third - cube? What does it mean? Very good question. Now you will have both squares and cubes.

Real life example #1

Let's start with the square or the second power of the number.

Imagine a square pool measuring one meter by one meter. The pool is at your dacha. It's hot and I really want to swim. But... the pool has no bottom! You need to cover the bottom of the pool with tiles. How many tiles do you need? In order to determine this, you need to know the bottom area of ​​the pool.

You can simply calculate by pointing your finger that the bottom of the pool consists of meter by meter cubes. If you have tiles one meter by one meter, you will need pieces. It's easy... But where have you seen such tiles? The tile will most likely be cm by cm. And then you will be tortured by “counting with your finger.” Then you have to multiply. So, on one side of the bottom of the pool we will fit tiles (pieces) and on the other, too, tiles. Multiply by and you get tiles ().

Did you notice that to determine the area of ​​the pool bottom we multiplied the same number by itself? What does it mean? Since we are multiplying the same number, we can use the “exponentiation” technique. (Of course, when you only have two numbers, you still need to multiply them or raise them to a power. But if you have a lot of them, then raising them to a power is much easier and there are also fewer errors in calculations. For the Unified State Exam, this is very important).
So, thirty to the second power will be (). Or we can say that thirty squared will be. In other words, the second power of a number can always be represented as a square. And vice versa, if you see a square, it is ALWAYS the second power of some number. A square is an image of the second power of a number.

Real life example #2

Here's a task for you: count how many squares there are on the chessboard using the square of the number... On one side of the cells and on the other too. To count their number, you need to multiply eight by eight or... if you notice that chessboard- this is a square with a side, then you can square eight. You will get cells. () So?

Real life example #3

Now the cube or the third power of a number. The same pool. But now you need to find out how much water will have to be poured into this pool. You need to calculate the volume. (Volumes and liquids, by the way, are measured in cubic meters. Unexpected, right?) Draw a pool: a bottom measuring a meter and a depth of a meter and try to count how many cubes measuring a meter by a meter will fit into your pool.

Just point your finger and count! One, two, three, four...twenty-two, twenty-three...How many did you get? Not lost? Is it difficult to count with your finger? That's it! Take an example from mathematicians. They are lazy, so they noticed that in order to calculate the volume of the pool, you need to multiply its length, width and height by each other. In our case, the volume of the pool will be equal to cubes... Easier, right?

Now imagine how lazy and cunning mathematicians are if they simplified this too. We reduced everything to one action. They noticed that the length, width and height are equal and that the same number is multiplied by itself... What does this mean? This means you can take advantage of the degree. So, what you once counted with your finger, they do in one action: three cubed is equal. It is written like this: .

All that remains is remember the table of degrees. Unless, of course, you are as lazy and cunning as mathematicians. If you like to work hard and make mistakes, you can continue to count with your finger.

Well, to finally convince you that degrees were invented by quitters and cunning people to solve their own life problems, and not to create problems for you, here are a couple more examples from life.

Real life example #4

You have a million rubles. At the beginning of each year, for every million you make, you make another million. That is, every million you have doubles at the beginning of each year. How much money will you have in years? If you are sitting now and “counting with your finger,” then you are a very hardworking person and... stupid. But most likely you will give an answer in a couple of seconds, because you are smart! So, in the first year - two multiplied by two... in the second year - what happened, by two more, in the third year... Stop! You noticed that the number is multiplied by itself times. So two to the fifth power is a million! Now imagine that you have a competition and the one who can count the fastest will get these millions... It’s worth remembering the powers of numbers, don’t you think?

Real life example #5

You have a million. At the beginning of each year, for every million you make, you earn two more. Great isn't it? Every million is tripled. How much money will you have in a year? Let's count. The first year - multiply by, then the result by another... It’s already boring, because you already understood everything: three is multiplied by itself times. So to the fourth power it is equal to a million. You just have to remember that three to the fourth power is or.

Now you know that by raising a number to a power you will make your life a lot easier. Let's take a further look at what you can do with degrees and what you need to know about them.

Terms and concepts... so as not to get confused

So, first, let's define the concepts. Do you think what is an exponent? It's very simple - it's the number that is "at the top" of the power of the number. Not scientific, but clear and easy to remember...

Well, at the same time, what such a degree basis? Even simpler - this is the number that is located below, at the base.

Here's a drawing for good measure.

Well in general view, in order to generalize and better remember... A degree with a base “ ” and an exponent “ ” is read as “to the degree” and is written as follows:

Power of a number with natural exponent

You probably already guessed: because the exponent is a natural number. Yes, but what is it natural number? Elementary! Natural numbers are those numbers that are used in counting when listing objects: one, two, three... When we count objects, we do not say: “minus five,” “minus six,” “minus seven.” We also do not say: “one third”, or “zero point five”. These are not natural numbers. What numbers do you think these are?

Numbers like “minus five”, “minus six”, “minus seven” refer to whole numbers. In general, integers include all natural numbers, numbers opposite to natural numbers (that is, taken with a minus sign), and number. Zero is easy to understand - it is when there is nothing. What do negative (“minus”) numbers mean? But they were invented primarily to indicate debts: if you have a balance on your phone in rubles, this means that you owe the operator rubles.

All fractions are rational numbers. How did they arise, do you think? Very simple. Several thousand years ago, our ancestors discovered that they lacked natural numbers to measure length, weight, area, etc. And they came up with rational numbers... Interesting, isn't it?

There are also irrational numbers. What are these numbers? In short, endless decimal. For example, if you divide the circumference of a circle by its diameter, you get an irrational number.

Resume:

Let us define the concept of a degree whose exponent is a natural number (i.e., integer and positive).

1. Any number to the first power is equal to itself:
2. To square a number means to multiply it by itself:
3. To cube a number means to multiply it by itself three times:

Definition. Raising a number to a natural power means multiplying the number by itself times:
.

Properties of degrees

Where did these properties come from? I'll show you now.

Let's see: what is it And ?

By definition:

How many multipliers are there in total?

It’s very simple: we added multipliers to the factors, and the result is multipliers.

But by definition, this is a power of a number with an exponent, that is: , which is what needed to be proven.

Example: Simplify the expression.

Solution:

Example: Simplify the expression.

Solution: It is important to note that in our rule Necessarily there must be the same reasons!
Therefore, we combine the powers with the base, but it remains a separate factor:

only for the product of powers!

Under no circumstances can you write that.

2. that's it th power of a number

Just as with the previous property, let us turn to the definition of degree:

It turns out that the expression is multiplied by itself times, that is, according to the definition, this is the th power of the number:

In essence, this can be called “taking the indicator out of brackets.” But you can never do this in total:

Let's remember the abbreviated multiplication formulas: how many times did we want to write?

But this is not true, after all.

Power with negative base

Up to this point, we have only discussed what the exponent should be.

But what should be the basis?

In powers of natural indicator the basis may be any number. Indeed, we can multiply any numbers by each other, be they positive, negative, or even.

Let's think about which signs ("" or "") will have powers of positive and negative numbers?

For example, is the number positive or negative? A? ? With the first one, everything is clear: no matter how many positive numbers we multiply by each other, the result will be positive.

But the negative ones are a little more interesting. We remember the simple rule from 6th grade: “minus for minus gives a plus.” That is, or. But if we multiply by, it works.

Determine for yourself what sign the following expressions will have:

1)	2)	3)
4)	5)	6)

Did you manage?

Here are the answers: In the first four examples, I hope everything is clear? We simply look at the base and exponent and apply the appropriate rule.

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In example 5) everything is also not as scary as it seems: after all, it doesn’t matter what the base is equal to - the degree is even, which means the result will always be positive.

Well, except when the base is zero. The base is not equal, is it? Obviously not, since (because).

Example 6) is no longer so simple!

6 examples to practice

Analysis of the solution 6 examples

If we ignore the eighth power, what do we see here? Let's remember the 7th grade program. So, do you remember? This is the formula for abbreviated multiplication, namely the difference of squares! We get:

Let's look carefully at the denominator. It looks a lot like one of the numerator factors, but what's wrong? The order of the terms is wrong. If they were reversed, the rule could apply.

But how to do this? It turns out that it’s very easy: the even degree of the denominator helps us here.

Magically the terms changed places. This “phenomenon” applies to any expression to an even degree: we can easily change the signs in parentheses.

But it's important to remember: all signs change at the same time!

Let's go back to the example:

And again the formula:

Whole we call the natural numbers, their opposites (that is, taken with the " " sign) and the number.

positive integer, and it is no different from natural, then everything looks exactly like in the previous section.

Now let's look at new cases. Let's start with an indicator equal to.

Any number to the zero power is equal to one:

As always, let us ask ourselves: why is this so?

Let's consider some degree with a base. Take, for example, and multiply by:

So, we multiplied the number by, and we got the same thing as it was - . What number should you multiply by so that nothing changes? That's right, on. Means.

We can do the same with an arbitrary number:

Let's repeat the rule:

Any number to the zero power is equal to one.

But there are exceptions to many rules. And here it is also there - this is a number (as a base).

On the one hand, it must be equal to any degree - no matter how much you multiply zero by itself, you will still get zero, this is clear. But on the other hand, like any number to the zero power, it must be equal. So how much of this is true? The mathematicians decided not to get involved and refused to raise zero to the zero power. That is, now we cannot not only divide by zero, but also raise it to the zero power.

Let's move on. In addition to natural numbers and numbers, integers also include negative numbers. To understand what a negative power is, let’s do as last time: multiply some normal number by the same number to a negative power:

From here it’s easy to express what you’re looking for:

Now let’s extend the resulting rule to an arbitrary degree:

So, let's formulate a rule:

A number with a negative power is the reciprocal of the same number with a positive power. But at the same time The base cannot be null:(because you can’t divide by).

Let's summarize:

I. The expression is not defined in the case. If, then.

II. Any number to the zero power is equal to one: .

III. A number not equal to zero to a negative power is the inverse of the same number to a positive power: .

Tasks for independent solution:

Well, as usual, examples for independent solutions:

Analysis of problems for independent solution:

I know, I know, the numbers are scary, but on the Unified State Exam you have to be prepared for anything! Solve these examples or analyze their solutions if you couldn’t solve them and you will learn to cope with them easily in the exam!

Let's continue to expand the range of numbers “suitable” as an exponent.

Now let's consider rational numbers. What numbers are called rational?

Answer: everything that can be represented as a fraction, where and are integers, and.

To understand what it is "fractional degree", consider the fraction:

Let's raise both sides of the equation to a power:

Now let's remember the rule about "degree to degree":

What number must be raised to a power to get?

This formulation is the definition of the root of the th degree.

Let me remind you: the root of the th power of a number () is a number that, when raised to a power, is equal to.

That is, the root of the th power is the inverse operation of raising to a power: .

It turns out that. Obviously this special case can be expanded: .

Now we add the numerator: what is it? The answer is easy to obtain using the power-to-power rule:

But can the base be any number? After all, the root cannot be extracted from all numbers.

None!

Let us remember the rule: any number raised to an even power is a positive number. That is, it is impossible to extract even roots from negative numbers!

This means that such numbers cannot be raised to fractional power with an even denominator, that is, the expression does not make sense.

What about the expression?

But here a problem arises.

The number can be represented in the form of other, reducible fractions, for example, or.

And it turns out that it exists, but does not exist, but these are just two different entries the same number.

Or another example: once, then you can write it down. But if we write down the indicator differently, we will again get into trouble: (that is, we got a completely different result!).

To avoid such paradoxes, we consider only positive base exponent with fractional exponent.

So if:

• — natural number;
• - integer;

Examples:

Degrees with rational indicator very useful for converting expressions with roots, for example:

5 examples to practice

Analysis of 5 examples for training

Well, now comes the hardest part. Now we'll figure it out degree with irrational exponent.

All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception

After all, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with natural, integer and rational exponents, each time we created a certain “image”, “analogy”, or description in more familiar terms.

For example, a degree with a natural exponent is a number multiplied by itself several times;

...number to the zeroth power- this is, as it were, a number multiplied by itself once, that is, they have not yet begun to multiply it, which means that the number itself has not even appeared yet - therefore the result is only a certain “blank number”, namely a number;

...negative integer degree- it’s as if some “reverse process” had occurred, that is, the number was not multiplied by itself, but divided.

By the way, in science a degree with a complex exponent is often used, that is, the exponent is not even a real number.

But at school we don’t think about such difficulties; you will have the opportunity to comprehend these new concepts at the institute.

WHERE WE ARE SURE YOU WILL GO! (if you learn to solve such examples :))

For example:

Decide for yourself:

Analysis of solutions:

1. Let's start with the usual rule for raising a power to a power:

Now look at the indicator. Doesn't he remind you of anything? Let us recall the formula for abbreviated multiplication of difference of squares:

In this case,

It turns out that:

Answer: .

2. We reduce fractions in exponents to the same form: either both decimals or both ordinary ones. We get, for example:

Answer: 16

3. Nothing special, let’s use it normal properties degrees:

ADVANCED LEVEL

Determination of degree

A degree is an expression of the form: , where:

• — degree base;
• - exponent.

Degree with natural indicator (n = 1, 2, 3,...)

Raising a number to the natural power n means multiplying the number by itself times:

Degree with an integer exponent (0, ±1, ±2,...)

If the exponent is positive integer number:

Construction to the zero degree:

The expression is indefinite, because, on the one hand, to any degree is this, and on the other hand, any number to the th degree is this.

If the exponent is negative integer number:

(because you can’t divide by).

Once again about zeros: the expression is not defined in the case. If, then.

Examples:

Power with rational exponent

• — natural number;
• - integer;

Examples:

Properties of degrees

To make it easier to solve problems, let’s try to understand: where did these properties come from? Let's prove them.

Let's see: what is and?

By definition:

So, on the right side of this expression we get the following product:

But by definition it is a power of a number with an exponent, that is:

Q.E.D.

Example : Simplify the expression.

Solution : .

Example : Simplify the expression.

Solution : It is important to note that in our rule Necessarily there must be the same reasons. Therefore, we combine the powers with the base, but it remains a separate factor:

Another important note: this rule - only for product of powers!

Under no circumstances can you write that.

Just as with the previous property, let us turn to the definition of degree:

Let's regroup this work like this:

It turns out that the expression is multiplied by itself times, that is, according to the definition, this is the th power of the number:

In essence, this can be called “taking the indicator out of brackets.” But you can never do this in total: !

Let's remember the abbreviated multiplication formulas: how many times did we want to write? But this is not true, after all.

Power with a negative base.

Up to this point we have only discussed what it should be like indicator degrees. But what should be the basis? In powers of natural indicator the basis may be any number .

Indeed, we can multiply any numbers by each other, be they positive, negative, or even. Let's think about which signs ("" or "") will have powers of positive and negative numbers?

For example, is the number positive or negative? A? ?

With the first one, everything is clear: no matter how many positive numbers we multiply by each other, the result will be positive.

But the negative ones are a little more interesting. We remember the simple rule from 6th grade: “minus for minus gives a plus.” That is, or. But if we multiply by (), we get - .

And so on ad infinitum: with each subsequent multiplication the sign will change. We can formulate the following simple rules:

1. even degree, - number positive.
2. Negative number, built in odd degree, - number negative.
3. Positive number to any degree is a positive number.
4. Zero to any power is equal to zero.

Determine for yourself what sign the following expressions will have:

1.	2.	3.
4.	5.	6.

Did you manage? Here are the answers:

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In the first four examples, I hope everything is clear? We simply look at the base and exponent and apply the appropriate rule.

In example 5) everything is also not as scary as it seems: after all, it doesn’t matter what the base is equal to - the degree is even, which means the result will always be positive. Well, except when the base is zero. The base is not equal, is it? Obviously not, since (because).

Example 6) is no longer so simple. Here you need to find out which is less: or? If we remember that, it becomes clear that, which means the base is less than zero. That is, we apply rule 2: the result will be negative.

And again we use the definition of degree:

Everything is as usual - we write down the definition of degrees and divide them by each other, divide them into pairs and get:

Before we look at the last rule, let's solve a few examples.

Calculate the expressions:

Solutions :

If we ignore the eighth power, what do we see here? Let's remember the 7th grade program. So, do you remember? This is the formula for abbreviated multiplication, namely the difference of squares!

We get:

Let's look carefully at the denominator. It looks a lot like one of the numerator factors, but what's wrong? The order of the terms is wrong. If they were reversed, rule 3 could apply. But how? It turns out that it’s very easy: the even degree of the denominator helps us here.

If you multiply it by, nothing changes, right? But now it turns out like this:

Magically the terms changed places. This “phenomenon” applies to any expression to an even degree: we can easily change the signs in parentheses. But it's important to remember: All signs change at the same time! You can’t replace it with by changing only one disadvantage we don’t like!

Let's go back to the example:

And again the formula:

So now the last rule:

How will we prove it? Of course, as usual: let’s expand on the concept of degree and simplify it:

Well, now let's open the brackets. How many letters are there in total? times by multipliers - what does this remind you of? This is nothing more than a definition of an operation multiplication: There were only multipliers there. That is, this, by definition, is a power of a number with an exponent:

Example:

Degree with irrational exponent

In addition to information about degrees for the average level, we will analyze the degree with an irrational exponent. All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception - after all, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are integers (that is, irrational numbers are all real numbers except rational numbers).

When studying degrees with natural, integer and rational exponents, each time we created a certain “image”, “analogy”, or description in more familiar terms. For example, a degree with a natural exponent is a number multiplied by itself several times; a number to the zero power is, as it were, a number multiplied by itself times, that is, they have not yet begun to multiply it, which means that the number itself has not even appeared yet - therefore the result is only a certain “blank number”, namely a number; a degree with an integer negative exponent - it’s as if some “reverse process” had occurred, that is, the number was not multiplied by itself, but divided.

It is extremely difficult to imagine a degree with an irrational exponent (just as it is difficult to imagine a 4-dimensional space). It is rather a purely mathematical object that mathematicians created to extend the concept of degree to the entire space of numbers.

By the way, in science a degree with a complex exponent is often used, that is, the exponent is not even a real number. But at school we don’t think about such difficulties; you will have the opportunity to comprehend these new concepts at the institute.

So what do we do if we see an irrational exponent? We are trying our best to get rid of it! :)

For example:

Decide for yourself:

1)

2)

3)

Answers:

1. Let's remember the difference of squares formula. Answer: .
2. We reduce the fractions to the same form: either both decimals or both ordinary ones. We get, for example: .
3. Nothing special, we use the usual properties of degrees:

SUMMARY OF THE SECTION AND BASIC FORMULAS

Degree called an expression of the form: , where:

Degree with an integer exponent

a degree whose exponent is a natural number (i.e., integer and positive).

Power with rational exponent

degree, the exponent of which is negative and fractional numbers.

Degree with irrational exponent

a degree whose exponent is an infinite decimal fraction or root.

Properties of degrees

Features of degrees.

• Negative number raised to even degree, - number positive.
• Negative number raised to odd degree, - number negative.
• A positive number to any degree is a positive number.
• Zero is equal to any power.
• Any number to the zero power is equal.

NOW YOU HAVE THE WORD...

How do you like the article? Write below in the comments whether you liked it or not.

Tell us about your experience using degree properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck on your exams!

The power is used to simplify the operation of multiplying a number by itself. For example, instead of writing, you can write 4 5 (\displaystyle 4^(5))(an explanation for this transition is given in the first section of this article). Degrees make it easier to write long or complex expressions or equations; powers are also easily added and subtracted, resulting in a simplified expression or equation (for example, 4 2 ∗ 4 3 = 4 5 (\displaystyle 4^(2)*4^(3)=4^(5))).

Note: if you need to decide exponential equation(in such an equation the unknown is in the exponent), read.

Steps

Solving simple problems with degrees

Multiply the base of the exponent by itself a number of times equal to the exponent. If you need to solve a power problem by hand, rewrite the power as a multiplication operation, where the base of the power is multiplied by itself. For example, given a degree 3 4 (\displaystyle 3^(4)). In this case, the base of power 3 must be multiplied by itself 4 times: 3 ∗ 3 ∗ 3 ∗ 3 (\displaystyle 3*3*3*3). Here are other examples:

First, multiply the first two numbers. For example, 4 5 (\displaystyle 4^(5)) = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4*4*4*4*4). Don't worry - the calculation process is not as complicated as it seems at first glance. First multiply the first two fours and then replace them with the result. Like this:

• 4 5 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=4*4*4*4*4)
  • 4 ∗ 4 = 16 (\displaystyle 4*4=16)

1. Multiply the result (16 in our example) by the next number. Each subsequent result will increase proportionally. In our example, multiply 16 by 4. Like this:

   • 4 5 = 16 ∗ 4 ∗ 4 ∗ 4 (\displaystyle 4^(5)=16*4*4*4)
     • 16 ∗ 4 = 64 (\displaystyle 16*4=64)
   • 4 5 = 64 ∗ 4 ∗ 4 (\displaystyle 4^(5)=64*4*4)
     • 64 ∗ 4 = 256 (\displaystyle 64*4=256)
   • 4 5 = 256 ∗ 4 (\displaystyle 4^(5)=256*4)
     • 256 ∗ 4 = 1024 (\displaystyle 256*4=1024)
   • Continue multiplying the result of the first two numbers by the next number until you get your final answer. To do this, multiply the first two numbers, and then multiply the resulting result by the next number in the sequence. This method is valid for any degree. In our example you should get: 4 5 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4 = 1024 (\displaystyle 4^(5)=4*4*4*4*4=1024) .

2. Solve the following problems. Check your answer using a calculator.

   • 8 2 (\displaystyle 8^(2))
   • 3 4 (\displaystyle 3^(4))
   • 10 7 (\displaystyle 10^(7))

3. On your calculator, look for the key labeled "exp" or " x n (\displaystyle x^(n))", or "^". Using this key you will raise a number to a power. It is almost impossible to calculate a degree with a large indicator manually (for example, the degree 9 15 (\displaystyle 9^(15))), but the calculator can easily cope with this task. In Windows 7, the standard calculator can be switched to engineering mode; To do this, click “View” -> “Engineering”. To switch to normal mode, click “View” -> “Normal”.

   • Check your answer using search engine(Google or Yandex). Using the "^" key on your computer keyboard, enter the expression into the search engine, which will instantly display the correct answer (and possibly suggest similar expressions for you to study).

   Addition, subtraction, multiplication of powers

   1. You can add and subtract powers only if they have the same base. If you need to add powers with the same bases and exponents, then you can replace the addition operation with the multiplication operation. For example, given the expression 4 5 + 4 5 (\displaystyle 4^(5)+4^(5)). Remember that the degree 4 5 (\displaystyle 4^(5)) can be represented in the form 1 ∗ 4 5 (\displaystyle 1*4^(5)); Thus, 4 5 + 4 5 = 1 ∗ 4 5 + 1 ∗ 4 5 = 2 ∗ 4 5 (\displaystyle 4^(5)+4^(5)=1*4^(5)+1*4^(5) =2*4^(5))(where 1 +1 =2). That is, count the number of similar degrees, and then multiply that degree and this number. In our example, raise 4 to the fifth power, and then multiply the resulting result by 2. Remember that the addition operation can be replaced by the multiplication operation, for example, 3 + 3 = 2 ∗ 3 (\displaystyle 3+3=2*3). Here are other examples:

      • 3 2 + 3 2 = 2 ∗ 3 2 (\displaystyle 3^(2)+3^(2)=2*3^(2))
      • 4 5 + 4 5 + 4 5 = 3 ∗ 4 5 (\displaystyle 4^(5)+4^(5)+4^(5)=3*4^(5))
      • 4 5 − 4 5 + 2 = 2 (\displaystyle 4^(5)-4^(5)+2=2)
      • 4 x 2 − 2 x 2 = 2 x 2 (\displaystyle 4x^(2)-2x^(2)=2x^(2))

   2. When multiplying powers with the same base, their exponents are added (the base does not change). For example, given the expression x 2 ∗ x 5 (\displaystyle x^(2)*x^(5)). In this case, you just need to add the indicators, leaving the base unchanged. Thus, x 2 ∗ x 5 = x 7 (\displaystyle x^(2)*x^(5)=x^(7)). Here is a visual explanation of this rule:

      When raising a power to a power, the exponents are multiplied. For example, a degree is given. Since exponents are multiplied, then (x 2) 5 = x 2 ∗ 5 = x 10 (\displaystyle (x^(2))^(5)=x^(2*5)=x^(10)). The point of this rule is that you are multiplying by powers (x 2) (\displaystyle (x^(2))) on itself five times. Like this:

      • (x 2) 5 (\displaystyle (x^(2))^(5))
      • (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)*x^( 2)*x^(2)*x^(2))
      • Since the base is the same, the exponents simply add up: (x 2) 5 = x 2 ∗ x 2 ∗ x 2 ∗ x 2 ∗ x 2 = x 10 (\displaystyle (x^(2))^(5)=x^(2)*x^(2)* x^(2)*x^(2)*x^(2)=x^(10))

   3. A power with a negative exponent should be converted to a fraction (reverse power). It doesn't matter if you don't know what it is reciprocal degree. If you are given a degree with a negative exponent, e.g. 3 − 2 (\displaystyle 3^(-2)), write this degree in the denominator of the fraction (put 1 in the numerator), and make the exponent positive. In our example: 1 3 2 (\displaystyle (\frac (1)(3^(2)))). Here are other examples:

      When dividing degrees with the same base, their exponents are subtracted (the base does not change). The division operation is the opposite of the multiplication operation. For example, given the expression 4 4 4 2 (\displaystyle (\frac (4^(4))(4^(2)))). Subtract the exponent in the denominator from the exponent in the numerator (do not change the base). Thus, 4 4 4 2 = 4 4 − 2 = 4 2 (\displaystyle (\frac (4^(4))(4^(2)))=4^(4-2)=4^(2)) = 16 .

      • The power in the denominator can be written as follows: 1 4 2 (\displaystyle (\frac (1)(4^(2)))) = 4 − 2 (\displaystyle 4^(-2)). Remember that a fraction is a number (power, expression) with a negative exponent.

   4. Below are some expressions that will help you learn how to solve problems with exponents. The expressions given cover the material presented in this section. To see the answer, simply select the empty space after the equals sign.

   Solving problems with fractional exponents

   A power with a fractional exponent (for example, ) is converted to a root operation. In our example: x 1 2 (\displaystyle x^(\frac (1)(2))) = x (\displaystyle (\sqrt (x))). It doesn't matter what number is in the denominator. fractional indicator degrees. For example, x 1 4 (\displaystyle x^(\frac (1)(4)))- is the fourth root of “x”, that is x 4 (\displaystyle (\sqrt[(4)](x))) .

   1. If the exponent is an improper fraction, then the exponent can be decomposed into two powers to simplify the solution of the problem. There is nothing complicated about this - just remember the rule of multiplying powers. For example, a degree is given. Convert such a power into a root whose power is equal to the denominator of the fractional exponent, and then raise this root to a power equal to the numerator of the fractional exponent. To do this, remember that 5 3 (\displaystyle (\frac (5)(3))) = (1 3) ∗ 5 (\displaystyle ((\frac (1)(3)))*5). In our example:

      • x 5 3 (\displaystyle x^(\frac (5)(3)))
      • x 1 3 = x 3 (\displaystyle x^(\frac (1)(3))=(\sqrt[(3)](x)))
      • x 5 3 = x 5 ∗ x 1 3 (\displaystyle x^(\frac (5)(3))=x^(5)*x^(\frac (1)(3))) = (x 3) 5 (\displaystyle ((\sqrt[(3)](x)))^(5))
   2. Some calculators have a button to calculate exponents (you must first enter the base, then press the button, and then enter the exponent). It is denoted as ^ or x^y.
   3. Remember that any number to the first power is equal to itself, for example, 4 1 = 4. (\displaystyle 4^(1)=4.) Moreover, any number multiplied or divided by one is equal to itself, e.g. 5 ∗ 1 = 5 (\displaystyle 5*1=5) And 5 / 1 = 5 (\displaystyle 5/1=5).
   4. Know that the power 0 0 does not exist (such a power has no solution). If you try to solve such a degree on a calculator or on a computer, you will receive an error. But remember that any number to the zero power is 1, for example, 4 0 = 1. (\displaystyle 4^(0)=1.)
   5. In higher mathematics, which operates with imaginary numbers: e a i x = c o s a x + i s i n a x (\displaystyle e^(a)ix=cosax+isinax), Where i = (− 1) (\displaystyle i=(\sqrt (())-1)); e is a constant approximately equal to 2.7; a is an arbitrary constant. The proof of this equality can be found in any textbook on higher mathematics.

   Warnings

   • As the exponent increases, its value increases greatly. So if the answer seems wrong to you, it may actually be correct. You can check this by plotting any exponential function eg 2 x .