A degree with a rational indicator

Power Function IV.

§ 71. Degrees with zero and negative indicators

In § 69, we proved (see theorem 2) that t\u003e P.

(a. =/= 0)

Quite naturally desire to extend this formula and in case t. < P . But then the number t - P. It will be either negative or equal to zero. A. We still spoke only about degrees with natural indicators. Thus, we are faced with the need to enter into consideration of real numbers with zero and negative indicators.

Definition 1. Any number but , not equal zero, to zero degree equal to one, that is, when but =/= 0

but 0 = 1. (1)

For example, (-13,7) 0 \u003d 1; π 0 \u003d 1; (√2) 0 \u003d 1. The number 0 of zero degree does not have, that is, the expression 0 0 is not defined.

Definition 2.. If a but \u003d / \u003d 0 and p - natural number, then

but - N. = 1 /a. n. (2)

i.e the degree of any number, unequal zero, with a whole negative indicator is the fraction, the numerator of which is a unit, and the denominator is the degree of the same number A, but with an indicator opposite to the indicator of a given degree.

For example,

By taking these definitions, it can be proved that a. \u003d / \u003d 0, formula

verne for any natural numbers t. and N. , not just for t\u003e P. . To prove, it is enough to limit ourselves to the consideration of two cases: t \u003d P. and t.< .п because case m\u003e N. Already considered in § 69.

Let be t \u003d P. ; then . It means that the left part of equality (3) is equal to 1. The right part is t \u003d P. becomes

but m - N. = but n - N. = but 0 .

But by definition but 0 \u003d 1. Thus, the right side of equality (3) is also equal to 1. Consequently, t \u003d P. Formula (3) is true.

Now suppose that t.< п . Sharing the numerator and denominator of the fraction on but m. We will get:

As p\u003e T. then. Therefore . Using the degree of degree with a negative indicator, you can record .

So, for As required to prove. Formula (3) is now proven for any natural numbers t. and p .

Comment. Negative indicators allow you to record fractions without denominators. For example,

1 / 3 = 3 - 1 ; 2 / 5 = 2 5 - one ; at all, A. / b. = and B. - 1

However, one should not think that with such an entry, the fractions turn into integers. For example, 3. - 1 There is the same fraction as 1/3, 2 5 - 1 - the same fraction as 2/5, and so on.

Exercises

529. Calculate:

530. Record without denominants:

1) 1 / 8 , 2) 1 / 625 ; 3) 10 / 17 ; 4) - 2 / 3

531. Data decimal fractions written in the form of integer expressions using negative indicators:

1) 0,01; 3) -0,00033; 5) -7,125;

2) 0,65; 4) -0,5; 6) 75,75.

3) - 33 10 - 5

First level

The degree and properties. Exhaustive Guide (2019)

Why are you needed? Where will they come to you? Why do you need to spend time on their study?

To find out all about degrees, what they need about how to use their knowledge in everyday life read this article.

And, of course, the knowledge of degrees will bring you closer to the successful surrender of OGE or the EGE and to enter the university of your dreams.

Let "S GO ... (drove!)

Important remark! If instead of formulas you see abracadabra, clean the cache. To do this, click Ctrl + F5 (on Windows) or CMD + R (on Mac).

FIRST LEVEL

The exercise is the same mathematical operation as addition, subtraction, multiplication or division.

Now I will explain all the human language on very simple examples. Pay attention. Examples of elementary, but explaining important things.

Let's start with addition.

There is nothing to explain here. You all know everything: we are eight people. Everyone has two bottles of cola. How much is the cola? Right - 16 bottles.

Now multiplication.

The same example with a cola can be recorded differently :. Mathematics - People cunning and lazy. They first notice some patterns, and then invent the way how 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 reception called multiplication. Agree, it is considered easier and faster than.

So, to read faster, easier and without mistakes, you just need to remember table multiplication. Of course, you can do everything more slowly, harder and mistakes! But…

Here is the multiplication table. Repeat.

And the other, more beautiful:

And what other tricks came up with lazy mathematicians? Right - erection.

Erection

If you need to multiply the number for yourself five times, then mathematics say that you need to build this number in the fifth degree. For example, . Mathematics remember that two in the fifth degree is. And they solve such tasks in the mind - faster, easier and without errors.

For this you need only remember what is highlighted in color in the table of degrees of numbers. Believe it, it will greatly facilitate your life.

By the way, why the second degree is called square numbers, and the third - cuba? What does it mean? Very good question. Now there will be to you and squares, and Cuba.

Example from life number 1

Let's start with a square or from a second degree of number.

Imagine a square pool of meter size on a meter. The pool is on your dacha. Heat and really want to swim. But ... Pool without the bottom! You need to store the bottom of the pool tiles. How much do you need tiles? In order to determine this, you need to find out the area of \u200b\u200bthe bottom of the pool.

You can simply calculate, with a finger, that the bottom of the pool consists of a meter cubes per meter. If you have a meter tile for meter, you will need to pieces. It's easy ... But where did you see such a tile? The tile is more likely to see for see and then "finger to count" torture. Then you have to multiply. So, on one side of the bottom of the pool, we fit tiles (pieces) and on the other too tiles. Multiplying on, you will get tiles ().

Did you notice that in order to determine the area of \u200b\u200bthe bottom of the pool, did we multiply the same number by yourself? What does it mean? This is multiplied by the same number, we can take advantage of the "erection of the extermination". (Of course, when you have only two numbers, multiply them or raise them into the degree. But if you have a lot of them, it is much easier to raise them in terms of calculations, too much less. For the exam, it is very important).
So thirty to the second degree will (). Or we can say that thirty in the square will be. In other words, the second degree of number can always be represented as a square. And on the contrary, if you see a square - it is always the second degree of some number. Square is the image of a second degree number.

Example from life number 2

Here is the task, count how many squares on a chessboard with a square of the number ... on one side of the cells and on the other too. To calculate their quantity, you need to multiply eight or ... If you note that the chessboard is a square of the side, then you can build eight per square. It turns out cells. () So?

Example from life number 3

Now a cube or the third degree of number. The same pool. But now you need to know how much water will have to fill in this pool. You need to count the volume. (Volumes and liquids, by the way, are measured in cubic meters. Suddenly, really?) Draw a pool: bottom of the meter size and a depth of meters and try to count how much cubes the size of the meter on the meter will enter your pool.

Right show your finger and count! Once, two, three, four ... twenty two, twenty three ... how much did it happen? Did not come down? Difficult to count your finger? So that! Take an example with mathematicians. They are lazy, therefore noticed that to calculate the volume of the pool, it is necessary to multiply each other in length, width and height. In our case, the volume of the pool will be equal to cubes ... it is easier for the truth?

And now imagine, as far as Mathematics are lazy and cunning, if they are simplified. Brought all to one action. They noticed that the length, width and height is equal to and that the same number varnims itself on itself ... And what does this mean? This means that you can take advantage of the degree. So, what did you think with your finger, they do in one action: three in Cuba is equal. This is written so :.

It remains only remember Table degrees. If you are, of course, the same lazy and cunning as mathematics. If you like to work a lot and make mistakes - you can continue to count your finger.

Well, to finally convince you that the degrees came up with Lodii and cunnies to solve their life problems, and not to create problems you, here's another couple of examples from life.

Example from life number 4

You have a million rubles. At the beginning of each year you earn every million another million. That is, every million will double at the beginning of each year. How much money will you have in the years? If you are sitting now and "you think your finger", then you are a very hardworking person and .. stupid. But most likely you will answer in a couple of seconds, because you are smart! So, in the first year - two multiplied two ... in the second year - what happened, another two, on the third year ... Stop! You noticed that the number multiplies itself. So, two in the fifth degree - a million! And now imagine that you have a competition and these million will receive the one who will find faster ... It is worth remembering the degree of numbers, what do you think?

Example from life number 5

You have a million. At the beginning of each year you earn each million two more. Great truth? Every million triples. How much money will you have after a year? Let's count. The first year is to multiply on, then the result is still on ... already boring, because you have already understood everything: three is multiplied by itself. Therefore, the fourth degree is equal to a million. It is only necessary to remember that three in the fourth degree is or.

Now you know that with the help of the erection of the number, you will greatly facilitate your life. Let's look next to what you can do with the degrees and what you need to know about them.

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

So, for starters, let's define the concepts. What do you think, what is the indicator of the degree? It is very simple - this is the number that is "at the top" of the degree of number. Not scientifically, but it is clear and easy to remember ...

Well, at the same time that such a foundation degree? Even easier - this is the number that is below, at the base.

Here is a drawing for loyalty.

Well, in general, to summarize and better remember ... The degree with the basis "" and the indicator "" is read as "to degree" and is written as follows:

The degree of number with a natural indicator

You already probably guessed: because the indicator is a natural number. Yes, but what is natural number? Elementary! Natural These are the numbers that are used in the account when listing items: one, two, three ... We, when we consider items, do not say: "Minus five", "minus six", "minus seven". We also do not say: "one third", or "zero of whole, five tenths." These are not natural numbers. And what these numbers do you think?

Numbers like "minus five", "minus six", "minus seven" belong to whole numbers. In general, to whole numbers include all natural numbers, the numbers are opposite to natural (that is, taken with a minus sign), and the number. Zero understand easily - this is when nothing. And what do they mean negative ("minus") numbers? But they were invented primarily to designate debts: if you have a balance on the phone number, it means that you should operator rubles.

All sorts of fractions are rational numbers. How did they arise, what do you think? Very simple. Several thousand years ago, our ancestors found that they lack natural numbers to measure long, weight, square, etc. And they invented rational numbers... I wonder if it's true?

There are also irrational numbers. What is this number? If short, then an infinite decimal fraction. For example, if the circumference length is divided into its diameter, then the irrational number will be.

Summary:

We define the concept of degree, the indicator of which is a natural number (i.e., a whole and positive).

1. Any number to the first degree equally to itself:
2. Evaluate the number in the square - it means to multiply it by itself:
3. Evaluate the number in the cube - it means to multiply it by itself three times:

Definition. Evaluate the number in a natural degree - it means to multiply the number of all time for yourself:
.

Properties of degrees

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

Let's see: what is and ?

A-priory:

How many multipliers are here?

Very simple: we completed multipliers to multipliers, it turned out the factors.

But by definition, this is the degree of a number with an indicator, that is, that, that it was necessary to prove.

Example: Simplify the expression.

Decision:

Example: Simplify the expression.

Decision: It is important to notice that in our rule before Must be the same foundation!
Therefore, we combine degrees with the basis, but remains a separate multiplier:

only for the work of degrees!

In no case can not write that.

2. That is The degree of number

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

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is, there is a number of number:

In fact, this can be called "the indicator for brackets". But never can do it in the amount:

Recall the formula of abbreviated multiplication: how many times did we want to write?

But it is incorrect, because.

Negative

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

But what should be the basis?

In the degrees of S. natural indicator The base can be any number. And the truth, we can multiply each other any numbers, whether they are positive, negative, or even.

Let's think about what signs ("or" ") will have the degrees of positive and negative numbers?

For example, a positive or negative number? BUT? ? With the first, everything is clear: how many positive numbers we are not multiplied by each other, the result will be positive.

But with negative a little more interesting. After all, we remember a simple rule of grade 6: "Minus for minus gives a plus." That is, or. But if we multiply on, it will work out.

Determine independently, what sign the following expressions will have:

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

Cope?

Here are the answers: in the first four examples, I hope everything is understandable? Just look at the base and indicator, and apply the appropriate rule.

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

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

Well, with the exception of the case when the base is zero. The reason is not equal? Obviously no, because (because).

Example 6) is no longer so simple!

6 Examples for Training

Solutions of 6 examples

If you do not pay attention to the eighth degree, what do we see here? Remember the Grade 7 program. So, remembered? This is a formula for abbreviated multiplication, namely - the difference of squares! We get:

Carefully look at the denominator. He is very similar to one of the multipliers of the numerator, but what's wrong? Not the procedure of the terms. If they would change them in places, it would be possible to apply the rule.

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

Magically, the components changed in places. This "phenomenon" is applicable for any expression to an even degree: we can freely change signs in brackets.

But it is important to remember: all signs are changing at the same time.!

Let's go back for example:

And again the formula:

Integer We call natural numbers opposite to them (that is, taken with the sign "") and the number.

whole positive number, And it does not differ from natural, then everything looks exactly as in the previous section.

And now let's consider new cases. Let's start with an indicator equal to.

Any number to zero equal to one:

As always, we will ask me: why is it so?

Consider any degree with the basis. Take, for example, and domineering on:

So, we multiplied the number on, and got the same as it was. And for what number must be multiplied so that nothing has changed? That's right on. So.

We can do the same with an arbitrary number:

Repeat the rule:

Any number to zero equal to one.

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

On the one hand, it should be equal to any extent - how much zero itself is neither multiplied, still get zero, it is clear. But on the other hand, like any number to zero degree, should be equal. So what's the truth? Mathematics decided not to bind and refused to erect zero to zero. That is, now we can not only be divided into zero, but also to build it to zero.

Let's go further. In addition to natural numbers and numbers include negative numbers. To understand what a negative degree, we will do as last time: Domingly some normal number on the same to a negative degree:

From here it is already easy to express the desired:

Now we spread the resulting rule to an arbitrary degree:

So, we formulate the rule:

The number is a negative degree back to the same number to a positive degree. But at the same time the base can not be zero: (Because it is impossible to divide).

Let's summarize:

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

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

III. A number that is not equal to zero, to a negative degree back to the same number to a positive degree :.

Tasks for self solutions:

Well, as usual, examples for self solutions:

Task analysis for self solutions:

I know, I know, the numbers are terrible, but the exam should be ready for everything! Share these examples or scatter their decision, if I could not decide and you will learn to easily cope with them on the exam!

Continue expanding the circle of numbers, "suitable" as an indicator of the degree.

Now consider rational numbers. What numbers are called rational?

Answer: All that can be represented in the form of fractions, where and - integers, and.

To understand what is "Freight degree", Consider the fraction:

Erected both parts of the equation to the degree:

Now remember the rule about "Degree to degree":

What number should be taken to the degree to get?

This formulation is the definition of root degree.

Let me remind you: the root of the number () is called the number that is equal in the extermination.

That is, the root degree is an operation, reverse the exercise into the degree :.

Turns out that. Obviously, this particular case can be expanded :.

Now add a numerator: what is? The answer is easy to get with the help of the "degree to degree" rule:

But can the reason be any number? After all, the root can not be extracted from all numbers.

No one!

Remember the rule: any number erected into an even degree is the number positive. That is, to extract the roots of an even degree from negative numbers it is impossible!

This means that it is impossible to build such numbers into a fractional degree with an even denominator, that is, the expression does not make sense.

What about expression?

But there is a problem.

The number can be represented in the form of DRGIH, reduced fractions, for example, or.

And it turns out that there is, but does not exist, but it's just two different records of the same number.

Or another example: once, then you can write. But it is worthwhile to write to us in a different way, and again we get a nuisance: (that is, they received a completely different result!).

To avoid similar paradoxes, we consider only a positive foundation of degree with fractional indicator.

So, if:

• - natural number;
• - integer;

Examples:

The degrees with the rational indicator are very useful for converting expressions with roots, for example:

5 examples for training

Analysis of 5 examples for training

Well, now - the most difficult. Now we will understand irrational.

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

After all, by definition, irrational numbers are numbers that cannot be represented in the form of a fraction, where and - integers (that is, irrational numbers are all valid numbers except rational).

When studying degrees with natural, whole and rational indicator, we each time constituted a certain "image", "analogy", or a description in more familiar terms.

For example, a natural figure is a number, several times multiplied by itself;

...zero - this is how the number multiplied by itself once, that is, it has not yet begun to multiply, it means that the number itself has not even appeared - therefore the result is only a certain "billet number", namely the number;

...degree with a whole negative indicator "It seemed to have occurred a certain" reverse process ", that is, the number was not multiplied by itself, but Deli.

By the way, in science is often used with a complex indicator, that is, the indicator is not even a valid number.

But at school we do not think about such difficulties, you will have the opportunity to comprehend these new concepts at the Institute.

Where we are sure you will do! (If you learn to solve such examples :))

For example:

Solim yourself:

Debris:

1. Let's start with the usual rules for the exercise rules for us:

Now look at the indicator. Doesn't he remind you of anything? Remember the formula of abbreviated multiplication. Square differences:

In this case,

Turns out that:

Answer: .

2. We bring the fraction in the indicators of degrees to the same form: either both decimal or both ordinary. We obtain, for example:

Answer: 16.

3. Nothing special, we use the usual properties of degrees:

ADVANCED LEVEL

Determination of degree

The degree is called the expression of the form: where:

• — degree basis;
• - Indicator.

The degree with the natural indicator (n \u003d 1, 2, 3, ...)

Build a natural degree n - it means multiplying the number for yourself once:

The degree with the integer (0, ± 1, ± 2, ...)

If an indicator of the degree is software positive number:

Construction in zero degree:

The expression is indefinite, because, on the one hand, to any extent, it is, and on the other - any number of in degree is.

If an indicator of the degree is a whole negative number:

(Because it is impossible to divide).

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

Examples:

Rational

• - 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? We prove them.

Let's see: What is what?

A-priory:

So, in the right part of this expression, such a work is obtained:

But by definition, this is the degree of a number with an indicator, that is:

Q.E.D.

Example : Simplify the expression.

Decision : .

Example : Simplify the expression.

Decision : It is important to notice that in our rule beforethere must be the same bases. Therefore, we combine degrees with the basis, but remains a separate multiplier:

Another important note: this is a rule - only for the work of degrees!

In no case to the nerve to write that.

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

We regroup this work like this:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is - by the degree of number:

In fact, this can be called "the indicator for brackets". But never can do this in the amount:!

Recall the formula of abbreviated multiplication: how many times did we want to write? But it is incorrect, because.

Degree with a negative basis.

Up to this point, we only discussed what should be indicator degree. But what should be the basis? In the degrees of S. natural indicator The base can be any number .

And the truth, we can multiply each other any numbers, whether they are positive, negative, or even. Let's think about what signs ("or" ") will have the degrees of positive and negative numbers?

For example, a positive or negative number? BUT? ?

With the first, everything is clear: how many positive numbers we are not multiplied by each other, the result will be positive.

But with negative a little more interesting. After all, we remember a simple rule of grade 6: "Minus for minus gives a plus." That is, or. But if we will multiply on (), it turns out.

And so to infinity: each time the next multiplication will change the sign. Simple rules can be formulated:

1. even degree - number positive.
2. Negative number erected into odd degree - number negative.
3. A positive number to either degree is the number positive.
4. Zero to any degree is zero.

Determine independently, what sign the following expressions will have:

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

Cope? Here are the answers:

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

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

In example 5), everything is also not as scary, as it seems: it doesn't matter what is equal to the base - the degree is even, which means that the result will always be positive. Well, with the exception of the case when the base is zero. The reason is not equal? Obviously no, because (because).

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

And again we use the degree of degree:

All as usual - write down the definition of degrees and, divide them to each other, divide on the pairs and get:

Before you disassemble the last rule, we solve several examples.

Calculated expressions:

Solutions :

If you do not pay attention to the eighth degree, what do we see here? Remember the Grade 7 program. So, remembered? This is a formula for abbreviated multiplication, namely - the difference of squares!

We get:

Carefully look at the denominator. He is very similar to one of the multipliers of the numerator, but what's wrong? Not the procedure of the terms. If they were swapped in places, it would be possible to apply the rule 3. But how to do it? It turns out very easy: the even degree of denominator helps us.

If you draw it on, nothing will change, right? But now it turns out the following:

Magically, the components changed in places. This "phenomenon" is applicable for any expression to an even degree: we can freely change signs in brackets. But it is important to remember: all signs are changing at the same time!You can not replace on, changing only one disagreeable minus!

Let's go back for example:

And again the formula:

So now the last rule:

How will we prove? Of course, as usual: I will reveal the concept of degree and simplifies:

Well, now I will reveal brackets. How much will the letters get? Once on multipliers - what does it remind? It is nothing but the definition of the operation multiplication: In total there were factors. That is, it is, by definition, the degree of number with the indicator:

Example:

Irrational

In addition to information about degrees for the average level, we will analyze the degree with the irrational indicator. All the rules and properties of degrees here are exactly the same as for a degree with a rational indicator, with the exception - after all, by definition, irrational numbers are numbers that cannot be submitted in the form of a fraction, where - the integers (i.e., irrational numbers are All valid numbers besides rational).

When studying degrees with natural, whole and rational indicator, we each time constituted a certain "image", "analogy", or a description in more familiar terms. For example, a natural figure is a number, several times multiplied by itself; The number in zero degree is somehow the number multiplied by itself once, that is, it has not yet begun to multiply, it means that the number itself has not even appeared - therefore, only a certain "billet", namely, is the result; The degree with a whole negative indicator is as if a certain "reverse process" occurred, that is, the number was not multiplied by itself, but divided.

Imagine the degree with an irrational indicator is extremely difficult (just as it is difficult to submit a 4-dimensional space). It is rather a purely mathematical object that mathematics created to expand the concept of degree to the entire space of numbers.

By the way, in science is often used with a complex indicator, that is, the indicator is not even a valid number. But at school we do not 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 rate? We are trying to get rid of it with all the might! :)

For example:

Solim yourself:

1)

2)

3)

Answers:

1. We remember the formula the difference of squares. Answer:.
2. We give the fraction to the same form: either both decimal, or both ordinary. We get, for example:.
3. Nothing special, we use the usual properties of degrees:

Summary of section and basic formulas

Degree called the expression of the form: where:

Integer

the degree, the indicator of which is a natural number (i.e., a whole and positive).

Rational

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

Irrational

the degree, the indicator of which is an infinite decimal fraction or root.

Properties of degrees

Features of degrees.

• Negative number erected into even degree - number positive.
• Negative number erected into odd degree - number negative.
• A positive number to either degree is the number positive.
• Zero to any degree is equal.
• Any number to zero equal.

Now you need a word ...

How do you need an article? Write down in the comments like or not.

Tell me about your experience in using the properties of degrees.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck on the exams!

Answers:

No name

if we consider that a ^ x \u003d e ^ x * ln (a), it turns out that the same 0 ^ 0 \u003d 1 (limit, at x-\u003e 0)
Although the answer is "uncertainty" also acceptable

Zero in mathematics is not emptiness, this number is very close to "nothing", just like infinity only on the turn

Sewage:
0 ^ 0 \u003d 0 ^ (a-a) \u003d 0 ^ a * 0 ^ (- a) \u003d 0 ^ A / 0 ^ a \u003d 0/0
In this case, we divide on zero, and this operation over a field of real numbers is not defined.

6 years ago

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What will zero be equal if it is taken to zero?

Why is the number to degree 0 equal to 1? There is a rule that any number, in addition to zero, erected into a zero degree will be equal to one: 20 \u003d 1; 1.50 \u003d 1; 100000 \u003d 1 however why is it so? When the number is erected into a ratio with a natural indicator, it is meant that it is multiplied by itself as many times as the indicator of the degree: 43 \u003d 4 × 4 × 4; 26 \u003d 2 × 2 × 2 × 2 × 2 × 2 When the indicator of the degree is 1, then there is only one multiplier during the construction (if there can be no factor in general), and therefore the construction result is equal to the ground: 181 \u003d 18; (-3.4) 1 \u003d -3.4 But how in such a case be with a zero indicator? What is multiplied by? Let's try to go differently. It is known that if two degrees have the same bases, but different indicators, then the base can be left in the same, and the indicators are either folded with each other (if the degree is multiplied), or deduct the divider indicator from the dividery indicator (if divided): 32 × 31 \u003d 32 + 1 \u003d 33 \u003d 3 × 3 × 3 \u003d 27 45 ÷ 43 \u003d 45-3 \u003d 42 \u003d 4 × 4 \u003d 16 And now we consider such an example: 82 ÷ 82 \u003d 82-2 \u003d 80 \u003d? What if we do not use the property of degrees with the same base and make calculations in order of their follows: 82 ÷ 82 \u003d 64 ÷ 64 \u003d 1 So we got a cherished unit. Thus, the zero indicator of the extent seems to say that the number is not multiplied by itself, but is divided by itself. And hence it becomes clear why the expression 00 does not make sense. After all, it is impossible to divide on 0. It can be reasoned differently. If there is, for example, multiplication of degrees 52 × 50 \u003d 52 + 0 \u003d 52, then it follows that 52 was multiplied by 1. Consequently, 50 \u003d 1.

From the properties of degrees: a ^ n / a ^ m \u003d a ^ (nm) if n \u003d m, the result will be the unit except naturally a \u003d 0, in this case (since zero to any extent will be zero) would have a division on zero, Therefore, 0 ^ 0 does not exist

Account in different languages

Nizhny names from 0 to 9 in popular world languages.

Language	0	1	2	3	4	5	6	7	8	9
English	zero.	one.	two.	three.	four.	five.	six.	seven.	eight	nine
Bulgarian	nula	edino	two	three	chetsiri.	pet	pole	sem	axis	net
Hungarian	nulla	egy.	kettõ	három.	négy.	Öt.	hat.	hét.	nYOLC.	kilenc.
Dutch	nul.	een.	twee.	dRIE	vier	vijf.	zes.	zeven.	acht.	negen.
Danish	nul.	en	to.	tre.	fire	fem.	sEKS.	sYV.	oTTE	ni.
Spanish	cero.	uno.	dOS.	tres.	cuatro.	cinco.	seis	siete.	ocho.	nueve.
Italian	zero.	uno.	due.	tre.	quattro.	cinque	sEI	sette	oTTO	nove.
Lithuanian	nulis.	vienas.	du	trys.	keturi.	penki.	SHIP	septyni.	aðtuoni.	devyni.
German	nULL	eIN.	zwei.	drei.	vier	fünf.	sechs.	sieben.	acht.	neun.
Russian	zero	one	two	three	four	five	six	seven	eight	nine
Polish	zero.	jeden.	dWA.	tRZY.	cztery.	pIêæ.	sze¶æ.	siedem.	osiem.	dziewiêæ.
Portuguese	um.	dois	três.	quatro.	cinco.	seis	seTe	oito.	nove.
French	zéro.	un	dEUX	trois	quatre.	cINQ.	six.	sept.	huit.	neuf.
Czech	nula.	jedna	dVA	tøi.	èTyøi.	pìt.	¹est	sedm.	oSM.	devìt.
Swedish	noll	ett	tVA.	tre.	fyra.	fem.	sex	sJU.	aTTA	niO.
Estonian	nULL	Üks.	kaks.	kolm.	neli.	viis.	kuus.	sEITSE.	kaheksa.	Üheksa.

Negative and zero

Zero, negative and fractional degree

Zero indicator

Evaluate this number to some degree means to repeat it in a factory as many times as units in an indicator of the degree.

According to this definition, the expression: a. 0 does not make sense. But that the rule of dividing the degrees of the same number so that the value of the divider is equal to the division indicator, the definition was introduced:

The zero degree of any number will be equal to one.

Negative indicator

Expression a -M., in itself does not make sense. But for the rule of dividing the degrees of the same number, and in the case when the divider indicator is larger than the definite indicator, the definition was introduced:

Example 1. If this number consists of 5 hundred, 7 tens, 2 units and 9 hundredths, then it can be depicted as follows:

5 × 10 2 + 7 × 10 1 + 2 × 10 0 + 0 × 10 -1 + 9 × 10 -2 \u003d 572.09

Example 2. If this number consists of a dozens, b units, with tenths and D thousands of it, it can be portrayed as follows:

a. × 10 1 + b. × 10 0 + c. × 10 -1 + d. × 10 -3.

Actions on degrees with negative indicators

When multiplying the degrees of the same number, the indicators are folded.

When dividing the degrees of the same number, the divider indicator is deducted from the division.

To be taken into the degree of work, it is enough to build in this degree every fact separately:

To construct a fraction, it is enough to build this degree separately both members of the fraci:

When erecting a degree to another degree, the indicators of degrees are variable.

Fractional indicator

If a k. not there is a multiple n., then expression: does not make sense. But that the rule of extracting the root from the extent took place at any value of the indicator of the degree, the definition was introduced:

Thanks to the introduction of a new symbol, the root extraction can always be replaced by the exercise.

Actions on degrees with fractional indicators

Actions on degrees with fractional indicators are performed according to the same rules set for integer indicators.

In the proof of this situation, we will first assume that members of fractions: and serving indicators of degrees are positive.

In particular n. or q. can be equal to one.

When multiplying the degrees of the same number, fractional indicators fold:

When dividing the degrees of the same number with fractional indicators, the divider indicator is deducted from the divide indicator:

To raise a degree to another degree in case of fractional indicators, it is sufficient to multiply the degrees:

To extract the root of fractional degree, it is quite enough to divide the degree to the root rate:

The rules of action are applicable not only to positive fractional indicators, but also to negative.

There is a rule that any number besides zero, erected to zero degree will be equal to one:
2 0 = 1; 1.5 0 = 1; 10 000 0 = 1
However, why is it so?
When the number is erected into a ratio with a natural figure, it is meant that it is multiplied by itself as many times as an indicator:
4 3 \u003d 4 × 4 × 4; 2 6 \u003d 2 × 2 × 2 × 2 × 2 x 2
When the indicator of the degree is 1, then during the construction there is only one multiplier (if there can be no factor in general), and therefore the construction result is equal to the ground:
18 1 = 18;(-3.4)^1 = -3.4
But how, in this case, with the zero indicator? What is multiplied by?
Let's try to go differently.

Why is the number to degree 0 equal to 1?

It is known that if two degrees have the same bases, but different indicators, then the base can be left in the same, and the indicators are either folded with each other (if the degree is multiplied), or the degree of the divider indicator from the dividery indicator (if degrees are divided):
3 2 × 3 1 \u003d 3 ^ (2 + 1) \u003d 3 3 \u003d 3 × 3 × 3 \u003d 27
4 5 ÷ 4 3 \u003d 4 ^ (5-3) \u003d 4 2 \u003d 4 × 4 \u003d 16
And now consider such an example:
8 2 ÷ 8 2 \u003d 8 ^ (2-2) \u003d 8 0 \u003d?
What if we do not use the property of degrees with the same basis and produce calculations in order of their following:
8 2 ÷ 8 2 \u003d 64 ÷ 64 \u003d 1
So we got a cherished unit. Thus, the zero indicator of the extent seems to say that the number is not multiplied by itself, but is divided by itself.
And hence it becomes clear why the expression 0 0 does not make sense. After all, it is impossible to divide 0.

There is a rule that any number besides zero, erected to zero degree will be equal to one:
20 = 1; 1.50 = 1; 100000 = 1

However, why is it so?

When the number is erected into a ratio with a natural figure, it is meant that it is multiplied by itself as many times as an indicator:
43 = 4...

0 0

In the algebra, the construction of a zero degree is common. What is the degree 0? What numbers can be erected into zero degree, and which no?

Definition.

Any number to zero degree, with the exception of zero, equal to one:

Thus, whatever the number is either degree 0, the result will always be the same - one.

And 1 to degrees 0, and 2 to degree 0, and any other number is a whole, fractional, positive, negative, rational, irrational - when the zero degree gives a unit.

The only exception is zero.

Zero in zero degree is not determined, such an expression does not make sense.

That is, any number can be erected into zero degree except zero.

If, when simplifying expressions with degrees, the number is obtained in a zero degree, it can be replaced by one:

If with ...

0 0

As part of the school program, it is believed that the value of the expression $% 0 ^ 0 $% is not defined.

From the point of view of modern mathematics, it is convenient to assume that $% 0 ^ 0 \u003d 1 $%. The idea here is next. Let there be a product of $% n $% of numbers of the type $% p_n \u003d x_1x_2 \\ ldots x_n $%. For all $% n \\ ge2 $%, the equality of $% p_n \u003d x_1x_2 \\ ldots x_n \u003d (x_1x_2 \\ ldots x_ (n - 1)) x_n \u003d p_ (n - 1) x_n $% is performed. It is convenient to consider this equality having a meaning and at $% n \u003d 1 $%, believing $% p_0 \u003d 1 $%. The logic here is this: Calculating the works, we first take 1, and then the dominant consistently on $% x_1 $%, $% x_2 $%, ..., $% x_n $%. It is this algorithm that is used when the works are being written when the program is written. If for some reason the logs did not happen, then the work remained equal to one.

In other words, it is convenient to consider having a meaning of such a thing as a "work of 0 submarines", considering it by definition to be equal to 1. In this case, you can also speak about the "empty work". If we are some number of households on this ...

0 0

Zero - it is zero. Roughly speaking, any degree of number is a piece of unity to the degree of times this is the number. Two in the third, let's say, it is 1 * 2 * 2 * 2, two in minus the first - 1/2. And then it is necessary so that there is no hole in the transition from positive degrees to a negative and vice versa.

x ^ n * x ^ (- n) \u003d 1 \u003d x ^ (n-n) \u003d x ^ 0

that's the whole point.

just and understand, thanks

x ^ 0 \u003d (x ^ 1) * (x ^ (- 1)) \u003d (1 / x) * (x / 1) \u003d 1

it is necessary for example, simply then to certain formulas that are valid for positive indicators - for example, X ^ n * x ^ m \u003d x ^ (M + N) - were still valid.
The same applies to the way and the definition of a negative degree as well as rational (i.e., for example 5 to grade 3/4)

\u003e And why do you need it at all?
For example, in statistics and the theorer are often played with zero by degrees.

And negative degrees do not interfere with you?
...

0 0

We continue to consider the properties of degrees, take for example, 16: 8 \u003d 2. Since 16 \u003d 24, and 8 \u003d 23, therefore, the division can be written in exponential form as 24: 23 \u003d 2, but if we deduct the exhibitors, then 24: 23 \u003d 21. Thus, we have to recognize that 2 and 21 are the same, therefore, 21 \u003d 2.

The same rule applies to any other exponential number, therefore, it is possible to formulate a rule in general form:

any number erected to the first degree remains unchanged.

This conclusion may have led you in amazement. You can still somehow understand the meaning of expression 21 \u003d 2, although the expression "one number two multiplied by itself" sounds rather strange. But the expression 20 means "not two numbers, ...

0 0

Definitions:

1. Zero degree

Any number other than zero, erected to zero degree equal to one. Zero to zero degree

2. Natural degree other than zero

Any number X, erected in the natural degree n, different from zero, equal to multiply n numbers x

3.1 The root of the natural extent other than zero

The root of the natural degree n, different from zero, from any positive number X is such a positive number Y, which, when erected into a degree n, gives the initial number x

3.2 The root of the odd natural degree

The root of the odd natural degree n from any number x is such a number Y, which, when erected into a degree n, gives the initial number x

3.3 root of any natural degree as a fractional degree

Extraction of the root of any natural degree n, different from zero, from any number x is the same thing that the construction of this number X into fractional degree 1 / N

0 0

Hello, dear Russel!

With the introduction of the concept, there is such an entry: "Expression value A ^ 0 \u003d 1"! It goes due to the logical concept of degree and neither a little different!
It commemorated when a young man is trying to get to the essence! But there are things that should just be perceived as granted!
You can design a new mathematics only when you study already open in centuries ago!
Of course, if we exclude that you are "not from the world of this" and you are given much more than the remaining sinful to us!

Note: Anna Mishevoy has an attempt to prove not proven! Also coming!
But there is one big "but" - in its proof there is no essential element: the case of dividing to zero!

Look at yourself what happens: 0 ^ 1/0 ^ 1 \u003d 0/0 !!!

But it is impossible to divide to zero!

Please, please carefully!

With the lot of best wishes and happiness of cash ...

0 0