Negative number to the power of zero. Raising to a power of zero - zero in different languages

First level

Degree and its properties. Comprehensive Guide (2019)

Why are degrees needed? Where do you need them? Why do you need to spend time studying them?

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

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

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

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

FIRST LEVEL

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

Now I will explain everything in human language using very simple examples. Be careful. 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. Each has two bottles of cola. How much cola? That's right - 16 bottles.

Now multiplication.

The same example with cola can be written in a different way: . Mathematicians are cunning and lazy people. They first notice some patterns, and then come up with a way to “count” them faster. In our case, they noticed that each of the eight people had the same number of bottles of cola 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, harder and with mistakes! But…

Here is the multiplication table. Repeat.

And another, prettier one:

And what other tricky counting tricks did lazy mathematicians come up with? Correctly - 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 this number to the fifth power. For example, . Mathematicians remember that two to the fifth power is. And they solve such problems in their mind - faster, easier and without errors.

To do this, you only need remember what is highlighted in color in the table of powers of numbers. Believe me, it will make your life much easier.

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

Real life example #1

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

Imagine a square pool measuring meters by meters. The pool is in your backyard. It's hot and I really want to swim. But ... a pool without a bottom! It is necessary 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 area of ​​the bottom of the pool.

You can simply count by poking your finger that the bottom of the pool consists of cubes meter by meter. If your tiles are meter by meter, you will need pieces. It's easy... But where did you see such a tile? The tile will rather be cm by cm. And then you will be tormented 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. Multiplying by, you get tiles ().

Did you notice that we multiplied the same number by itself to determine the area of ​​the bottom of the pool? What does it mean? Since the same number is multiplied, we can use the exponentiation technique. (Of course, when you have only two numbers, you still need to multiply them or raise them to a power. But if you have a lot of them, then raising to a power is much easier and there are also fewer errors in the calculations. For the exam, this is very important).
So, thirty to the second degree will be (). Or you 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 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 a chessboard is a square with a side, then you can square eight. 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 one meter in size and a meter deep and try to calculate how many cubes measuring a meter by a meter will enter your pool.

Just point your finger and count! One, two, three, four…twenty-two, twenty-three… How much did it turn out? Didn't get lost? Is it difficult to count with your finger? So that! 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 make that too easy. Reduced everything to one action. They noticed that the length, width and height are equal and that the same number is multiplied by itself ... And what does this mean? This means that you can use the degree. So, what you once counted with a finger, they do in one action: three in a cube is equal. It is written like this:

Remains only memorize 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 keep counting with your finger.

Well, in order to finally convince you that degrees were invented by loafers and cunning people to solve their 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, you earn another million for every million. That is, each of your million at the beginning of each year doubles. How much money will you have in years? If you are now sitting 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 times two ... in the second year - what happened, by two more, in the third year ... Stop! You noticed that the number is multiplied by itself once. So two to the fifth power is a million! Now imagine that you have a competition and the one who calculates faster will get these millions ... Is it worth remembering the degrees of numbers, what do you think?

Real life example #5

You have a million. At the beginning of each year, you earn two more for every million. It's great right? 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 the fourth power is a million. You just need 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 much 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. What do you think, what is exponent? It's very simple - this is 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 base of degree? Even simpler is the number that is at the bottom, at the base.

Here's a picture for you to be sure.

Well, in general terms, in order to generalize and remember better ... A degree with a base "" and an indicator "" is read as "in the degree" and is written as follows:

Power of a number with a natural exponent

You probably already guessed: because the exponent is a natural number. Yes, but what is natural number? Elementary! Natural numbers are those that are used in counting when listing items: one, two, three ... When we count items, we don’t say: “minus five”, “minus six”, “minus seven”. We don't say "one third" or "zero point five tenths" either. These are not natural numbers. What do you think these numbers 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 a number. Zero is easy to understand - this is when there is nothing. And what do negative ("minus") numbers mean? But they were invented primarily to denote 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 come about, do you think? Very simple. Several thousand years ago, our ancestors discovered that they did not have enough 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, an infinite decimal fraction. For example, if you divide the circumference of a circle by its diameter, then you get an irrational number.

Summary:

Let's define the concept of degree, the exponent of which is a natural number (that is, integer and positive).

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

Definition. To raise a number to a natural power is to multiply the number by itself times:
.

Degree properties

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

Let's see what is and ?

By definition:

How many multipliers are there in total?

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

But by definition, this is the degree of a number with an exponent, that is: , which was required to be proved.

Example: Simplify the expression.

Solution:

Example: Simplify the expression.

Solution: It is important to note that in our rule necessarily must be the same reason!
Therefore, we combine the degrees with the base, but remain a separate factor:

only for products of powers!

Under no circumstances should you write that.

2. that is -th power of a number

Just as with the previous property, let's 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 the th power of the number:

In fact, this can be called "bracketing the indicator". But you can never do this in total:

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

But that's not true, really.

Degree with a negative base

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

But what should be the basis?

In degrees from natural indicator the basis may be any number. Indeed, we can multiply any number by each other, whether they are positive, negative, or even.

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

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

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

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: it doesn’t matter what the base is equal to - the degree is even, which means that the result will always be positive.

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

Example 6) is no longer so simple!

6 practice examples

Analysis of the solution 6 examples

If we do not pay attention to the eighth degree, what do we see here? Let's take a look at the 7th grade program. So, remember? This is the abbreviated multiplication formula, namely the difference of squares! We get:

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

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

The terms have magically changed places. This "phenomenon" applies to any expression to an even degree: we can freely change the signs in brackets.

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 name 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, we ask ourselves: why is this so?

Consider some power with a base. Take, for example, and multiply by:

So, we multiplied the number by, and got the same as it was -. What number must be multiplied 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 still get zero, this is clear. But on the other hand, like any number to the zero degree, it must be equal. So what is the truth of this? Mathematicians decided not to get involved and refused to raise zero to the zero power. That is, now we can not only divide by zero, but also raise it to the zero power.

Let's go further. In addition to natural numbers and numbers, integers include negative numbers. To understand what a negative degree is, let's do the same as last time: we multiply some normal number by the same in a negative degree:

From here it is already easy to express the desired:

Now we extend the resulting rule to an arbitrary degree:

So, let's formulate the rule:

A number to a negative power is the inverse of the same number to a positive power. But at the same time base cannot be null:(because it is impossible to divide).

Let's summarize:

I. Expression is not defined in case. If, then.

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

III. A number that is 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 an independent solution:

Analysis of tasks for independent solution:

I know, I know, the numbers are scary, but at the exam you have to be ready for anything! Solve these examples or analyze their solution if you couldn't solve it and you will learn how to easily deal with them in the exam!

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

Now consider rational numbers. What numbers are called rational?

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

To understand what is "fractional degree" Let's consider a fraction:

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

Now remember the rule "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.

That is, the root of the th degree is the inverse operation of exponentiation: .

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

Now add the numerator: what is it? The answer is easy to get with the power-to-power rule:

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

None!

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

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

What about expression?

But here a problem arises.

The number can be represented as other, reduced fractions, for example, or.

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

Or another example: once, then you can write it down. But as soon as we write the indicator in a different way, we again get trouble: (that is, we got a completely different result!).

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

So if:

  • - natural number;
  • is an integer;

Examples:

Powers with a rational exponent are very useful for transforming expressions with roots, for example:

5 practice examples

Analysis of 5 examples for training

Well, now - the most difficult. Now we will analyze degree with an irrational exponent.

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

Indeed, 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 a natural, integer and rational indicator, each time we made up a certain “image”, “analogy”, or description in more familiar terms.

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

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

...negative integer exponent- it’s as if a certain “reverse process” has taken place, that is, the number was not multiplied by itself, but divided.

By the way, science often uses a degree with a complex exponent, that is, an 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 how to solve such examples :))

For example:

Decide for yourself:

Analysis of solutions:

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

Now look at the score. Does he remind you of anything? We recall the formula for abbreviated multiplication of the difference of squares:

AT this case,

It turns out that:

Answer: .

2. We bring fractions in exponents to the same form: either both decimal or both ordinary. We get, for example:

Answer: 16

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

ADVANCED LEVEL

Definition of degree

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

  • base of degree;
  • - exponent.

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

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

Power with integer exponent (0, ±1, ±2,...)

If the exponent is positive integer number:

erection to zero power:

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 integer negative number:

(because it is impossible to divide).

One more time about nulls: the expression is not defined in the case. If, then.

Examples:

Degree with rational exponent

  • - natural number;
  • is an integer;

Examples:

Degree properties

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, the following product is obtained:

But by definition, this 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 must have the same basis. Therefore, we combine the degrees with the base, but remain a separate factor:

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

Under no circumstances should I write that.

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

Let's rearrange it like this:

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

In fact, this can be called "bracketing the indicator". But you can never do this in total:!

Let's recall the formulas for abbreviated multiplication: how many times did we want to write? But that's not true, really.

Power with a negative base.

Up to this point, we have discussed only what should be index degree. But what should be the basis? In degrees from natural indicator the basis may be any number .

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

For example, will the number be positive or negative? BUT? ?

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

But the negative ones are a little more interesting. After all, we remember a simple rule from the 6th grade: “a minus times a 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. You can formulate these simple rules:

  1. even degree, - number positive.
  2. Negative number raised to odd degree, - number negative.
  3. A positive number to any power 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: it doesn’t matter what the base is equal to - the degree is even, which means that the result will always be positive. Well, except when the base is zero. The base is not the same, is it? Obviously not, since (because).

Example 6) is no longer so simple. Here you need to find out which is less: or? If you remember that, it becomes clear that, which means that 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 into each other, divide them into pairs and get:

Before analyzing the last rule, let's solve a few examples.

Calculate the values ​​of expressions:

Solutions :

If we do not pay attention to the eighth degree, what do we see here? Let's take a look at the 7th grade program. So, remember? This is the abbreviated multiplication formula, namely the difference of squares!

We get:

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

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

The terms have magically changed places. This "phenomenon" applies to any expression to an even degree: we can freely change the signs in brackets. But it's important to remember: all signs change at the same time! It cannot be replaced by by changing only one objectionable minus to us!

Let's go back to the example:

And again the formula:

So now the last rule:

How are we going to prove it? Of course, as usual: let's expand the concept of degree and simplify:

Well, now let's open the brackets. How many letters will there be? times by multipliers - what does it look like? This is nothing but the definition of an operation multiplication: total there turned out to be multipliers. That is, it is, by definition, a power of a number with an exponent:

Example:

Degree with irrational exponent

In addition to information about the degrees for the average level, we will analyze the degree with an irrational indicator. 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 a natural, integer and rational indicator, each time we made up a certain “image”, “analogy”, or description in more familiar terms. For example, a natural exponent is a number multiplied by itself several times; a number to the zero degree is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared yet - therefore, the result is only a certain “preparation of a number”, namely a number; a degree with an integer negative indicator - it is as if a certain “reverse process” has 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). Rather, it is a purely mathematical object that mathematicians have created to extend the concept of a degree to the entire space of numbers.

By the way, science often uses a degree with a complex exponent, that is, an 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. Remember the difference of squares formula. Answer: .
  2. We bring fractions to the same form: either both decimals, or both ordinary ones. We get, for example: .
  3. Nothing special, we apply the usual properties of degrees:

SECTION SUMMARY AND BASIC FORMULA

Degree is called an expression of the form: , where:

Degree with integer exponent

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

Degree with rational exponent

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

Degree with irrational exponent

exponent whose exponent is an infinite decimal fraction or root.

Degree properties

Features of degrees.

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

NOW YOU HAVE A WORD...

How do you like the article? Let me know in the comments below if you liked it or not.

Tell us about your experience with the power properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck with your exams!

There is a rule that any number other than zero, raised to the power of zero, will be equal to one:
20 = 1; 1.50 = 1; 100000 = 1

However, why is this so?

When a number is raised to a power with a natural exponent, it means that it is multiplied by itself as many times as the exponent:
43 = 4...

0 0

In algebra, raising to a power of zero is common. What is degree 0? Which numbers can be raised to the zero power and which cannot?

Definition.

Any number to the power of zero, except for zero, is equal to one:

Thus, no matter what number is raised to the power of 0, the result will always be the same - one.

And 1 to the power of 0, and 2 to the power of 0, and any other number - integer, fractional, positive, negative, rational, irrational - when raised to the zero power, gives one.

The only exception is null.

Zero to the zero power is not defined, such an expression does not make sense.

That is, any number except zero can be raised to the zero power.

If, when simplifying an expression with powers, a number is obtained to the power of zero, it can be replaced by a unit:

If at...

0 0

Within the framework of the school curriculum, the value of the expression $%0^0$% is considered undefined.

From the point of view of modern mathematics, it is convenient to assume that $%0^0=1$%. The idea here is the following. Let there be a product of $%n$% numbers of the form $%p_n=x_1x_2\ldots x_n$%. For all $%n\ge2$% the equality $%p_n=x_1x_2\ldots x_n=(x_1x_2\ldots x_(n-1))x_n=p_(n-1)x_n$% is satisfied. It is convenient to consider this equality to be meaningful even for $%n=1$%, setting $%p_0=1$%. The logic here is as follows: when calculating products, we first take 1, and then multiply successively by $%x_1$%, $%x_2$%, ..., $%x_n$%. It is this algorithm that is used when finding works when programs are written. If, for some reason, multiplication did not occur, then the product remains equal to one.

In other words, it is convenient to consider such a concept as "the product of 0 factors" as having meaning, considering it equal to 1 by definition. In this case, one can also speak of an "empty product". If we multiply a number by this...

0 0

Zero - it is zero. Roughly speaking, any power of a number is the product of one and the exponent times that number. Two in the third, let's say it's 1*2*2*2, two in minus the first is 1/2. And then it is necessary that there be no hole in the transition from positive to negative powers and vice versa.

x^n * x^(-n) = 1 = x^(n-n) = x^0

that's the whole point.

simple and clear, thanks

x^0=(x^1)*(x^(-1))=(1/x)*(x/1)=1

it is necessary for example simply then that certain formulas that are valid for positive indicators - for example x ^ n * x ^ m = x ^ (m + n) - are still valid.
By the way, the same applies to the definition of a negative degree as well as a rational one (that is, for example, 5 to the power of 3/4)

> and why is it needed at all?
For example, in statistics and theory, one often plays with zero powers.

Do negative degrees bother you?
...

0 0

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

The same rule applies to any other exponential number, so the rule can be stated in a general way:

any number raised to the first power remains unchanged

This conclusion may have surprised you. You can still somehow understand the meaning of the expression 21=2, although the expression "one number two multiplied by itself" sounds rather strange. But the expression 20 means "not a single number two, ...

0 0

Degree definitions:

1. zero degree

Any non-zero number raised to the power of zero equals one. Zero to the power of zero is not defined

2. natural degree other than zero

Any number x raised to a natural power n, other than zero, is equal to the multiplication of n numbers x among themselves

3.1 root of an even natural degree other than zero

The root of an even natural power n, different from zero, from any positive number x is such a positive number y that, when raised to a power of n, gives the original number x

3.2 odd natural root

An odd natural root n of any number x is a number y that, when raised to a power of n, gives the original number x

3.3 the root of any natural power as a fractional power

Extracting the root of any natural power n other than zero from any number x is the same as raising this number x to the fractional power 1/n

0 0

Hello, dear RUSSEL!

When introducing the concept of degree, there is such a notation: » The value of the expression a^0 =1 » ! This goes by virtue of the logical concept of degree and nothing else!
It's commendable when a young man tries to get to the bottom of it! But there are things that should just be taken for granted!
You can construct new mathematics only when you study what was already discovered centuries ago!
Of course, if we exclude that you are "not of this world" and you have been given much more than the rest of us sinners!

Note: Anna Misheva made an attempt to prove the unprovable! Also commendable!
But there is one big "BUT" - the most important element is missing from its proof: The case of division by ZERO!

See for yourself what can happen: 0^1 / 0^1 = 0 / 0 !!!

But you can't divide by zero!

Please be more careful!

With a lot of best wishes and happiness in your personal life...

0 0

Answers:

No name

if we take into account that a^x=e^x*ln(a), then it turns out that 0^0=1 (limit, for x->0)
although the answer "uncertainty" is also acceptable

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

Write down:
0^0 = 0^(a-a) = 0^a * 0^(-a) = 0^a / 0^a = 0 / 0
It turns out in this case we divide by zero, and this operation on the field of real numbers is not defined.

6 years ago

RPI.su is the largest Russian-language database of questions and answers. Our project was implemented as a continuation of the popular service otvety.google.ru, which was closed and removed on April 30, 2015. We decided to resurrect the useful Google Answers service so that any person can publicly find out the answer to their question from the Internet community.

All questions added to the Google Answers site have been copied and saved here. The names of old users are also displayed in the form in which they previously existed. You only need to re-register to be able to ask questions, or answer others.

To contact us for any question ABOUT THE SITE (advertising, cooperation, feedback about the service), write to the mail [email protected] Only post all general questions on the site, they will not be answered by mail.

What is zero equal to when it is raised to the power of zero?

Why is a number to the power of 0 equal to 1? There is a rule that any number other than zero, raised to the power of zero, will be equal to one: 20 = 1; 1.50 = 1; 100000 = 1 However, why is this so? When a number is raised to a power with a natural exponent, it means that it is multiplied by itself as many times as the exponent: 43 = 4 × 4 × 4; 26 \u003d 2 × 2 × 2 × 2 × 2 × 2 When the exponent is 1, then there is only one factor during the construction (if we can talk about factors here at all), and therefore the result of the construction is equal to the base of the degree: 181 = 18; (–3.4)1 = –3.4 But what about the zero exponent in this case? What is multiplied by what? Let's try to go the other way. It is known that if two degrees have the same bases, but different indicators, then the base can be left the same, and the indicators can either be added to each other (if the degrees are multiplied), or subtract the divisor indicator from the dividend indicator (if the degrees are divisible): 32 × 31 = 32+1 = 33 = 3 × 3 × 3 = 27 45 ÷ 43 = 45–3 = 42 = 4 × 4 = 16 Now consider this example: 82 ÷ 82 = 82–2 = 80 = ? What if we do not use the property of degrees with the same base and make calculations in the order they follow: 82 ÷ 82 = 64 ÷ 64 = 1 So we got the coveted unit. Thus, the zero exponent, as it were, indicates that the number is not multiplied by itself, but is divided by itself. And from here it becomes clear why the expression 00 does not make sense. After all, you cannot divide by 0. You can argue differently. If there is, for example, a multiplication of powers 52 × 50 = 52 + 0 = 52, then it follows that 52 has been multiplied by 1. Therefore, 50 = 1.

From the properties of degrees: a^n / a^m = a^(n-m) if n=m, the result will be one, except of course a=0, in this case (since zero to any degree will be zero) division by zero would take place, so 0^0 doesn't exist

Account in different languages

Names of numerals from 0 to 9 in popular languages ​​of the world.

Language 0 1 2 3 4 5 6 7 8 9
English zero one two three four five six seven eight nine
Bulgarian zero one two three four pet pole sedem osem devet
Hungarian nulla egy ketto harom negy ot hat het nyolc kilenc
Dutch null een twee drie vier vijf zes zeven acht negen
Danish null en to tre fire fem sexes syv otte ni
Spanish cero uno dos tres cuatro cinco seis siete ocho new
Italian zero uno due tre quattro cinque sei sette otto new
Lithuanian nullis vienas du trys keturi penki reyi septini aðtuoni devyni
Deutsch null ein zwei drei vier funf 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 tracks quadro cinco seis sete oito new
French zero un deux trois square cinq six sept huit neuf
Czech nula jedna dva toi ityoi pit ¹est sedm osm devite
Swedish noll ett tva tre fyra fem sex sju atta nio
Estonian null uks kaks Kolm neli viis kuus seitse kaheksa uheksa

Negative and zero power of a number

Zero, negative and fractional powers

Zero indicator

To raise a given number to a certain power means to repeat it with a factor as many times as there are units in the exponent.

According to this definition, the expression: a 0 is meaningless. But in order for the rule of dividing the powers of the same number to have meaning even in the case when the divisor index is equal to the dividend index, the definition is introduced:

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

Negative indicator

Expression a-m, by itself is meaningless. But in order for the rule of dividing the powers of the same number to have meaning even in the case when the divisor index is greater than the dividend index, the definition is introduced:

Example 1. If a given number consists of 5 hundreds, 7 tens, 2 units and 9 hundredths, then it can be represented as follows:

5 × 10 2 + 7 × 10 1 + 2 × 10 0 + 0 × 10 -1 + 9 × 10 -2 = 572.09

Example 2. If a given number consists of a tens, b units, c tenths and d thousandths, then it can be represented as follows:

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

Actions on powers with negative exponents

When multiplying powers of the same number, the exponents are added together.

When dividing the powers of the same number, the divisor indicator is subtracted from the indicator of the dividend.

To raise a product to a power, it is enough to raise each factor separately to this power:

To raise a fraction to a power, it is enough to raise both terms of the fraction separately to this power:

When a power is raised to another power, the exponents are multiplied.


Fractional exponent

If a k is not a multiple n, then the expression: does not make sense. But in order for the rule of extracting the root from the degree to take place for any value of the exponent, the definition is introduced:

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

Actions on powers with fractional exponents

Actions on degrees with fractional exponents are performed according to the same rules that are established for integer exponents.

When proving this position, we will first assume that the terms of the fractions: and , serving as exponents, are positive.

In a particular case n or q may be equal to one.

When multiplying the powers of the same number, fractional indicators add up:


When dividing powers of the same number with fractional exponents, the divisor exponent is subtracted from the dividend exponent:

To raise a power to another power in the case of fractional exponents, it is enough to multiply the exponents:

To extract the root of a fractional exponent, it is enough to divide the exponent by the exponent of the root:

The rules of action apply not only to positive fractional figures, but also to negative.

There is a rule that any number other than zero, raised to the power of zero, will be equal to one:
2 0 = 1; 1.5 0 = 1; 10 000 0 = 1
However, why is this so?
When a number is raised to a power with a natural exponent, it means that it is multiplied by itself as many times as the exponent:
4 3 = 4×4×4; 2 6 = 2×2×2×2×2 x 2
When the exponent is 1, then there is only one factor during the construction (if we can talk about factors here at all), and therefore the result of the construction is equal to the base of the degree:
18 1 = 18;(-3.4)^1 = -3.4
But what about zero in this case? What is multiplied by what?
Let's try to go the other way.

Why is a number to the power of 0 equal to 1?

It is known that if two degrees have the same bases, but different indicators, then the base can be left the same, and the indicators can either be added to each other (if the degrees are multiplied), or subtract the divisor indicator from the dividend indicator (if the degrees are divisible):
3 2×3 1 = 3^(2+1) = 3 3 = 3×3×3 = 27
4 5 ÷ 4 3 = 4^(5−3) = 4 2 = 4×4 = 16
Now consider this example:
8 2 ÷ 8 2 = 8^(2−2) = 8 0 = ?
What if we do not use the property of powers with the same base and perform calculations in the order of their sequence:
8 2 ÷ 8 2 = 64 ÷ 64 = 1
So we got the coveted unit. Thus, the zero exponent, as it were, indicates that the number is not multiplied by itself, but is divided by itself.
And from here it becomes clear why the expression 0 0 does not make sense. After all, you can't divide by 0.

DEGREE WITH A RATIONAL INDICATOR,

POWER FUNCTION IV

§ 71. Degrees with zero and negative exponents

In § 69 we proved (see Theorem 2) that for t > n

(a =/= 0)

It is quite natural to want to extend this formula to the case when t < P . But then the number t - p will be either negative or zero. A. We have so far only talked about degrees with natural indicators. Thus, we are faced with the need to introduce into consideration the powers of real numbers with zero and negative exponents.

Definition 1. Any number a , not equal to zero, to the power of zero is equal to one, that is, when a =/= 0

a 0 = 1. (1)

For example, (-13.7) 0 = 1; π 0 = 1; (√2) 0 = 1. The number 0 has no zero degree, that is, the expression 0 0 is not defined.

Definition 2. If a a=/= 0 and P is a natural number, then

a - n = 1 /a n (2)

that is the degree of any number that is not equal to zero, with a negative integer exponent, is equal to a fraction whose numerator is one, and the denominator is the power of the same number a, but with an exponent opposite to the exponent of this exponent.

For example,

With these definitions in mind, it can be shown that a =/= 0, formula

true for any natural numbers t and n , and not only for t > n . To prove it, it suffices to consider only two cases: t = n and t< .п , since the case m > n already dealt with in § 69.

Let t = n ; then . Hence, the left side of equality (3) is equal to 1. The right side at t = n becomes

a m-n = a n - n = a 0 .

But by definition a 0 = 1. Thus, the right side of equality (3) is also equal to 1. Therefore, for t = n formula (3) is correct.

Now suppose that t< п . Dividing the numerator and denominator of a fraction by a m , we get:

Because n > t , then . That's why . Using the definition of a degree with a negative exponent, one can write .

So, at , which was to be proved. Formula (3) is now proved for any natural numbers t and P .

Comment. Negative exponents allow you to write fractions without denominators. For example,

1 / 3 = 3 - 1 ; 2 / 5 = 2 5 - one ; generally, a / b = a b - 1

However, one should not think that with such a notation, fractions turn into whole numbers. For example, 3 - 1 is the same fraction as 1/3, 2 5 - 1 is the same fraction as 2/5, etc.

Exercises

529. Calculate:

530. Write without denominators of a fraction:

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

531. Write these decimal fractions as 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