DEGREE WITH RATIONAL INDICATOR,

POWER FUNCTION IV

§ 71. Powers with zero and negative exponents

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

(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 equal to zero. A. We have so far only talked about degrees with natural exponents. Thus, we are faced with the need to introduce powers of real numbers with zero and negative exponents into consideration.

Definition 1. Any number A , Not equal to zero, to the zeroth power 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 does not have a zero degree, that is, the expression 0 0 is not defined.

Definition 2. If A=/= 0 and P - natural number, That

A - n = 1 /a n (2)

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

For example,

Having accepted these definitions, it can be proven that when a =/= 0, formula

true for any natural numbers T And n , and not just for t > p . To prove it, it is enough to limit ourselves to considering two cases: t = n And T< .п , since the case m > n already discussed in § 69.

Let t = n ; Then . Means, left side 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-hand side of equality (3) is also equal to 1. Therefore, when t = n formula (3) is correct.

Now suppose that T< п . Divide the numerator and denominator of the fraction by A m , we get:

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

So, when , which was what needed to be proven. Formula (3) has now been proven 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 - 1 ; at all, a / b = a b - 1

However, you should not think that with this 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 a fraction without denominators:

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

531. Write these decimal fractions in the form of whole expressions using negative exponents:

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

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

3) - 33 10 - 5

First 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 success passing the OGE or the Unified State Exam and admission to 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).

FIRST LEVEL

Raising to a power is the same mathematical operation like addition, subtraction, multiplication or division.

Now I'll explain everything human language 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 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 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 calculate 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. 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? 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 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,” it means you are very hardworking man 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. What 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 number c natural indicator

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?

Is there some more 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.

Summary:

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 will show you now.

Let's see: what is it And ?

A-priory:

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 degrees 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 that? 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.

whole positive number , 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 zero degree. 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 degree is, let's do as in last time: multiply some normal number to the same to a negative degree:

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 a 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 something happened “ reverse process", that is, the number was not multiplied by itself, but divided.

By the way, in science a degree with complex indicator, that is, the indicator is not even 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, we use the usual properties of 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?

A-priory:

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 index 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 degrees 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. A negative number, built in odd degree, - number negative.
3. A 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 you take it apart 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 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; 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's rather clean mathematical object, which 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 irrational indicator degrees? 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!

There is a rule that any number other than zero raised to the zero power 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 the zero power is common. What is degree 0? Which numbers can be raised to the zero power and which cannot?

Definition.

Any number to the zero power, except 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 zero.

Zero to the zeroth power is not defined, such an expression has no meaning.

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

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

If...

0 0

Within school curriculum The expression $%0^0$% is considered to be undefined.

From point of view 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 $%p_n=x_1x_2\ldots x_n=(x_1x_2\ldots x_(n-1))x_n=p_(n-1)x_n$% holds. It is convenient to consider this equality to be meaningful also for $%n=1$%, assuming $%p_0=1$%. The logic here is this: when calculating products, we first take 1, and then multiply sequentially by $%x_1$%, $%x_2$%, ..., $%x_n$%. This is the algorithm that is used to find products when programs are written. If for some reason the multiplications 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” to have meaning, considering it equal to 1 by definition. In this case, we can also talk about the “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 this number. Two in the third, let's say, is 1*2*2*2, two in the minus of the first is 1/2. And then it is necessary that there is no hole during the transition from positive degrees to negative and vice versa.

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

that's the whole point.

simple and clear, thank you

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

For example, you just need to have certain formulas that are valid for positive indicators- for example x^n*x^m=x^(m+n) - were 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 this even necessary?
For example, in statistics and theory they often play with zero powers.

A negative powers are they bothering you?
...

0 0

We continue to consider the properties of degrees, take for example 16:8 = 2. Since 16=24 and 8=23, therefore, 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 thing, therefore 21 = 2.

The same rule applies to any other exponential number, thus, the rule can be formulated in general form:

any number raised to the first power remains unchanged

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

0 0

Degree definitions:

1. zero degree

Any number other than zero raised to the zero power is equal to one. Zero to the zeroth power is undefined

2. natural degree other than zero

Any number x raised to a natural power n other than zero is equal to multiplying n numbers x together

3.1 even root natural degree, different from zero

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

3.2 root of odd natural degree

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

3.3 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 the following entry: “The value of the expression a^0 =1” ! This goes into effect logical concept degrees and nothing else!
It's commendable when a young man tries to get to the bottom of things! But there are some things that should simply be taken for granted!
You can construct new mathematics only when you have already studied open for centuries back!
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” - it is missing from her proof essential element: Case of division by ZERO!

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

But you CANNOT DIVIDE BY ZERO!

Please be more careful!

With mass 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, it is a number 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 that in this case we are dividing by zero, and this operation on the field of real numbers is not defined.

6 years ago

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What will zero be equal to if it is raised to the zero power?

Why does a number to the power of 0 equal 1? There is a rule that any number other than zero raised to the zero power 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 = 2 × 2 × 2 × 2 × 2 × 2 When the exponent is equal to 1, then during the construction there is only one factor (if we can talk about factors at all), and therefore the result of the construction equal to the base degrees: 181 = 18; (–3.4)1 = –3.4 But what about the zero indicator in this case? What is multiplied by what? Let's try to go a different 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 exponents can either be added to each other (if the powers are multiplied), or the exponent of the divisor can be subtracted from the exponent of the dividend (if the powers are divided): 32 × 31 = 32+1 = 33 = 3 × 3 × 3 = 27 45 ÷ 43 = 45–3 = 42 = 4 × 4 = 16 And now consider this example: 82 ÷ 82 = 82–2 = 80 = ? What if we don't use the property of powers with the same basis and let’s carry out the calculations in the order in which they appear: 82 ÷ 82 = 64 ÷ 64 = 1 So we got the treasured unit. Thus, the zero exponent seems to indicate that the number is not multiplied by itself, but divided by itself. And from here it becomes clear why the expression 00 does not make sense. After all, you can’t divide by 0. You can reason differently. If there is, for example, a multiplication of powers of 52 × 50 = 52+0 = 52, then it follows that 52 was multiplied by 1. Therefore, 50 = 1.

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

Accounting in different languages

Names of numerals from 0 to 9 on popular languages peace.

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 thing	two	three	four	pet	pole	we're getting ready	axes	devet
Hungarian	nulla	egy	kettõ	harom	négy	ot	hat	het	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	nullis	vienas	du	trys	keturi	penki	ðeði	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	zero	un	deux	trois	quatre	cinq	six	sept	huit	neuf
Czech	nula	jedna	dva	toi	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 powers of a number

Zero, negative and fractional powers

Zero indicator

Erect given number to some degree means to repeat it by a factor as many times as there are units in the exponent.

According to this definition, the expression: a 0 doesn't make sense. But for the rule of dividing powers of the same number to be valid even in the case when the exponent of the divisor equal to the indicator of the dividend, a definition has been introduced:

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

Negative indicator

Expression a -m, in itself has no meaning. But so that the rule for dividing powers of the same number is valid even in the case when the exponent of the divisor is greater than the exponent of the dividend, a definition has been introduced:

Example 1. If a given number consists of 5 hundreds, 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 = 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 add up.

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

If k is not a multiple of n, then the expression: makes no sense. But in order for the rule for extracting the root of a degree to take place for any value of the exponent, a definition has been introduced:

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

Actions on powers with fractional exponents

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

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

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

When multiplying powers of the same number, fractional exponents are added:

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

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 power, it is enough to divide the exponent by the exponent of the root:

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

There is a rule that any number other than zero raised to the zero power 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 equal to 1, then during the construction there is only one factor (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 the zero indicator in this case? What is multiplied by what?
Let's try to go a different way.

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

It is known that if two powers have the same bases, but different exponents, then the base can be left the same, and the exponents can either be added to each other (if the powers are multiplied), or the exponent of the divisor can be subtracted from the exponent of the dividend (if the powers 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 let's look at 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 carry out calculations in the order they appear:
8 2 ÷ 8 2 = 64 ÷ 64 = 1
So we received the coveted unit. Thus, the zero exponent seems to indicate that the number is not multiplied by itself, but divided by itself.
And from here it becomes clear why the expression 0 0 does not make sense. You can't divide by 0.