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
Exponentiation is a mathematical operation just 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 terms, in order to generalize and remember better... 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 for measuring 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, it's an infinite decimal fraction. For example, if the circumference of a circle is divided by its diameter, then we get irrational number.
Summary:
Let us define the concept of a degree whose exponent is a natural number (i.e., integer and positive).
- Any number to the first power is equal to itself:
- To square a number means to multiply it by itself:
- To cube a number means to multiply it by itself three times:
Definition. Raise the number to natural degree- means multiplying a 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.
positive integer, and it is no different from natural, then everything looks exactly like in the previous section.
Now let's look at new cases. Let's start with an indicator equal to.
Any number to the zero power is equal to one:
As always, let us ask ourselves: why is this so?
Let's consider some degree with a base. Take, for example, and multiply by:
So, we multiplied the number by, and we got the same thing as it was - . What number should you multiply by so that nothing changes? That's right, on. Means.
We can do the same with an arbitrary number:
Let's repeat the rule:
Any number to the zero power is equal to one.
But there are exceptions to many rules. And here it is also there - this is a number (as a base).
On the one hand, it must be equal to any degree - no matter how much you multiply zero by itself, you will still get zero, this is clear. But on the other hand, like any number to the zero power, it must be equal. So how much of this is true? The mathematicians decided not to get involved and refused to raise zero to 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 of degree c fractional indicator .
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;
...degree with integer negative indicator - 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:
- even degree, - number positive.
- Negative number raised to odd degree, - number negative.
- A positive number to any degree is a positive number.
- 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:
- Let's remember the difference of squares formula. Answer: .
- We reduce the fractions to the same form: either both decimals or both ordinary ones. We get, for example: .
- 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!
In this article we will figure out what it is degree of. Here we will give definitions of the power of a number, while we will consider in detail all possible exponents, starting with the natural exponent and ending with the irrational one. In the material you will find a lot of examples of degrees, covering all the subtleties that arise.
Page navigation.
Power with natural exponent, square of a number, cube of a number
Let's start with . Looking ahead, let's say that the definition of the power of a number a with natural exponent n is given for a, which we will call degree basis, and n, which we will call exponent. We also note that a degree with a natural exponent is determined through a product, so to understand the material below you need to have an understanding of multiplying numbers.
Definition.
Power of a number with natural exponent n is an expression of the form a n, the value of which is equal to the product of n factors, each of which is equal to a, that is, .
In particular, the power of a number a with exponent 1 is the number a itself, that is, a 1 =a.
It’s worth mentioning right away about the rules for reading degrees. Universal method reading the entry a n is: “a to the power of n”. In some cases, the following options are also acceptable: “a to the nth power” and “nth power of a”. For example, let's take the power 8 12, this is “eight to the power of twelve”, or “eight to the twelfth power”, or “twelfth power of eight”.
The second power of a number, as well as the third power of a number, have their own names. The second power of a number is called square the number, for example, 7 2 is read as “seven squared” or “the square of the number seven.” The third power of a number is called cubed numbers, for example, 5 3 can be read as “five cubed” or you can say “cube of the number 5”.
It's time to bring examples of degrees with natural exponents. Let's start with the degree 5 7, here 5 is the base of the degree, and 7 is the exponent. Let's give another example: 4.32 is the base, and the natural number 9 is the exponent (4.32) 9 .
Please note that in last example The base of the degree 4.32 is written in parentheses: to avoid discrepancies, we will put in brackets all bases of the degree that are different from natural numbers. As an example, we give the following degrees with natural exponents 
, their bases are not natural numbers, so they are written in parentheses. Well, for complete clarity, at this point we will show the difference contained in records of the form (−2) 3 and −2 3. The expression (−2) 3 is a power of −2 with a natural exponent of 3, and the expression −2 3 (it can be written as −(2 3) ) corresponds to the number, the value of the power 2 3 .
Note that there is a notation for the power of a number a with an exponent n of the form a^n. Moreover, if n is a multi-valued natural number, then the exponent is taken in brackets. For example, 4^9 is another notation for the power of 4 9 . And here are some more examples of writing degrees using the symbol “^”: 14^(21) , (−2,1)^(155) . In what follows, we will primarily use degree notation of the form a n .
One of the problems inverse to raising to a power with a natural exponent is the problem of finding the base of the power by known value degrees and known indicator. This task leads to .
It is known that many rational numbers consists of whole and fractional numbers, and each fractional number can be represented as positive or negative common fraction. We defined a degree with an integer exponent as previous paragraph, therefore, to complete the definition of a degree with a rational exponent, you need to give meaning to the degree of the number a with a fractional exponent m/n, where m is an integer and n is a natural number. Let's do it.
Let's consider a degree with a fractional exponent of the form . For the power-to-power property to remain valid, the equality must hold 
. If we take into account the resulting equality and how we determined , then it is logical to accept it provided that for given m, n and a the expression makes sense.
It is easy to check that for all properties of a degree with an integer exponent are valid (this was done in the section properties of a degree with a rational exponent).
The above reasoning allows us to make the following conclusion: if given m, n and a the expression makes sense, then the power of a with a fractional exponent m/n is called the nth root of a to the power of m.
This statement brings us close to the definition of a degree with a fractional exponent. All that remains is to describe at what m, n and a the expression makes sense. Depending on the restrictions placed on m, n and a, there are two main approaches.
The easiest way is to impose a constraint on a by taking a≥0 for positive m and a>0 for negative m (since for m≤0 the degree 0 of m is not defined). Then we get following definition degrees with a fractional exponent.
Definition.
Degree positive number a with fractional exponent m/n, where m is an integer and n is a natural number, is called the nth root of the number a to the power m, that is, .
The fractional power of zero is also determined with the only caveat that the indicator must be positive.
Definition.
Power of zero with fractional positive indicator m/n, where m is a positive integer and n is a natural number, is defined as 
.
When the degree is not determined, that is, the degree of the number zero with a fractional negative exponent does not make sense.
It should be noted that with this definition of a degree with a fractional exponent, there is one caveat: for some negative a and some m and n, the expression makes sense, and we discarded these cases by introducing the condition a≥0. For example, the entries make sense 
or , and the definition given above forces us to say that powers with a fractional exponent of the form 
do not make sense, since the base should not be negative.
Another approach to determining a degree with a fractional exponent m/n is to separately consider even and odd exponents of the root. This approach requires additional condition: the power of a number whose exponent is , is considered to be a power of a number whose exponent is the corresponding irreducible fraction(The importance of this condition will be explained below). That is, if m/n is an irreducible fraction, then for any natural number k the degree is first replaced by .
For even n and positive m, the expression makes sense for any non-negative a (even root of negative number does not make sense), for negative m the number a must still be different from zero (otherwise there will be division by zero). And for odd n and positive m, the number a can be any (the root of an odd degree is defined for any real number), and for negative m, the number a must be non-zero (so that there is no division by zero).
The above reasoning leads us to this definition of a degree with a fractional exponent.
Definition.
Let m/n be an irreducible fraction, m an integer, and n a natural number. For any reducible fraction, the degree is replaced by . The power of a number with an irreducible fractional exponent m/n is for
Let us explain why a degree with a reducible fractional exponent is first replaced by a degree with an irreducible exponent. If we simply defined the degree as , and did not make a reservation about the irreducibility of the fraction m/n, then we would be faced with situations similar to the following: since 6/10 = 3/5, then the equality must hold 
, But 
, A .
Degree formulas used in the process of reduction and simplification complex expressions, in solving equations and inequalities.
Number c is n-th power of a number a When:
Operations with degrees.
1. Multiplying powers of c the same basis their indicators add up:
a m·a n = a m + n .
2. When dividing degrees with the same base, their exponents are subtracted:
3. Power of the product of 2 or more factors is equal to the product of the powers of these factors:
(abc…) n = a n · b n · c n …
4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:
(a/b) n = a n /b n .
5. Raising a power to a power, the exponents are multiplied:
(a m) n = a m n .
Each formula above is true in the directions from left to right and vice versa.
For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.
Operations with roots.
1. The root of the product of several factors is equal to the product of the roots of these factors:
2. Root of attitude equal to the ratio dividend and divisor of roots:
3. When raising a root to a power, it is enough to raise the radical number to this power:
4. If you increase the degree of the root in n once and at the same time build into n th power is a radical number, then the value of the root will not change:
5. If you reduce the degree of the root in n extract the root at the same time n-th power of a radical number, then the value of the root will not change:
A degree with a negative exponent. The power of a certain number with a non-positive (integer) exponent is defined as one divided by the power of the same number with an exponent equal to absolute value non-positive indicator:
Formula a m:a n =a m - n can be used not only for m> n, but also with m< n.
For example. a4:a 7 = a 4 - 7 = a -3.
To formula a m:a n =a m - n became fair when m=n, the presence of zero degree is required.
A degree with a zero index. The power of any number, not equal to zero, with a zero exponent equals one.
For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.
Degree with a fractional exponent. To raise a real number A to the degree m/n, you need to extract the root n th degree of m-th power of this number A.
One of the main characteristics in algebra, and in all mathematics, is degree. Of course, in the 21st century, all calculations can be done on an online calculator, but it is better for brain development to learn how to do it yourself.
In this article we will look at the most important questions related to this definition. Namely, let’s understand what it is in general and what its main functions are, what properties there are in mathematics.
Let's look at examples of what the calculation looks like and what the basic formulas are. Let's look at the main types of quantities and how they differ from other functions.
Let's understand how to solve using this quantity various tasks. We will show with examples how to raise to the zero power, irrational, negative, etc.
Online exponentiation calculator
What is a power of a number
What is meant by the expression “raise a number to a power”?
The power n of a number is the product of factors of magnitude a n times in a row.
Mathematically it looks like this:
a n = a * a * a * …a n .
For example:
- 2 3 = 2 in the third degree. = 2 * 2 * 2 = 8;
- 4 2 = 4 to step. two = 4 * 4 = 16;
- 5 4 = 5 to step. four = 5 * 5 * 5 * 5 = 625;
- 10 5 = 10 in 5 steps. = 10 * 10 * 10 * 10 * 10 = 100000;
- 10 4 = 10 in 4 steps. = 10 * 10 * 10 * 10 = 10000.
Below is a table of squares and cubes from 1 to 10.
Table of degrees from 1 to 10
Below are the results of raising natural numbers to positive powers - “from 1 to 100”.
| Ch-lo | 2nd st. | 3rd stage |
| 1 | 1 | 1 |
| 2 | 4 | 8 |
| 3 | 9 | 27 |
| 4 | 16 | 64 |
| 5 | 25 | 125 |
| 6 | 36 | 216 |
| 7 | 49 | 343 |
| 8 | 64 | 512 |
| 9 | 81 | 279 |
| 10 | 100 | 1000 |
Properties of degrees
What is characteristic of such mathematical function? Let's look at the basic properties.
Scientists have established the following signs characteristic of all degrees:
- a n * a m = (a) (n+m) ;
- a n: a m = (a) (n-m) ;
- (a b) m =(a) (b*m) .
Let's check with examples:
2 3 * 2 2 = 8 * 4 = 32. On the other hand, 2 5 = 2 * 2 * 2 * 2 * 2 =32.
Similarly: 2 3: 2 2 = 8 / 4 =2. Otherwise 2 3-2 = 2 1 =2.
(2 3) 2 = 8 2 = 64. What if it’s different? 2 6 = 2 * 2 * 2 * 2 * 2 * 2 = 32 * 2 = 64.
As you can see, the rules work.
But what about with addition and subtraction? It's simple. Exponentiation is performed first, and then addition and subtraction.
Let's look at examples:
- 3 3 + 2 4 = 27 + 16 = 43;
- 5 2 – 3 2 = 25 – 9 = 16. Please note: the rule will not apply if you subtract first: (5 – 3) 2 = 2 2 = 4.
But in this case, you need to calculate the addition first, since there are actions in parentheses: (5 + 3) 3 = 8 3 = 512.
How to produce calculations in more difficult cases ? The order is the same:
- if there are brackets, you need to start with them;
- then exponentiation;
- then perform the operations of multiplication and division;
- after addition, subtraction.
Eat specific properties, not typical for all degrees:
- The nth root of a number a to the m degree will be written as: a m / n.
- When raising a fraction to a power: both the numerator and its denominator are subject to this procedure.
- When constructing a work different numbers to a power, the expression will correspond to the product of these numbers to the given power. That is: (a * b) n = a n * b n .
- When raising a number to a negative power, you need to divide 1 by a number in the same century, but with a “+” sign.
- If the denominator of a fraction is to a negative power, then this expression will be equal to the product of the numerator and the denominator to a positive power.
- Any number to the power 0 = 1, and to the power. 1 = to yourself.
These rules are important in some cases; we will consider them in more detail below.
Degree with a negative exponent
What to do when minus degree, i.e. when the indicator is negative?
Based on properties 4 and 5(see point above), it turns out:
A (- n) = 1 / A n, 5 (-2) = 1 / 5 2 = 1 / 25.
And vice versa:
1 / A (- n) = A n, 1 / 2 (-3) = 2 3 = 8.
What if it's a fraction?
(A / B) (- n) = (B / A) n, (3 / 5) (-2) = (5 / 3) 2 = 25 / 9.
Degree with natural indicator
It is understood as a degree with exponents equal to integers.
Things to remember:
A 0 = 1, 1 0 = 1; 2 0 = 1; 3.15 0 = 1; (-4) 0 = 1...etc.
A 1 = A, 1 1 = 1; 2 1 = 2; 3 1 = 3...etc.
In addition, if (-a) 2 n +2 , n=0, 1, 2...then the result will be with a “+” sign. If a negative number is raised to an odd power, then vice versa.
General properties and that's it specific signs, described above, are also characteristic of them.
Fractional degree
This type can be written as a scheme: A m / n. Read as: the nth root of the number A to the power m.
You can do whatever you want with a fractional indicator: reduce it, split it into parts, raise it to another power, etc.
Degree with irrational exponent
Let α be an irrational number and A ˃ 0.
To understand the essence of a degree with such an indicator, Let's look at different possible cases:
- A = 1. The result will be equal to 1. Since there is an axiom - 1 in all powers is equal to one;
А r 1 ˂ А α ˂ А r 2 , r 1 ˂ r 2 – rational numbers;
- 0˂А˂1.
In this case, it’s the other way around: A r 2 ˂ A α ˂ A r 1 under the same conditions as in the second paragraph.
For example, the exponent is the number π. It's rational.
r 1 – in this case equals 3;
r 2 – will be equal to 4.
Then, for A = 1, 1 π = 1.
A = 2, then 2 3 ˂ 2 π ˂ 2 4, 8 ˂ 2 π ˂ 16.
A = 1/2, then (½) 4 ˂ (½) π ˂ (½) 3, 1/16 ˂ (½) π ˂ 1/8.
Such degrees are characterized by all mathematical operations and specific properties described above.
Conclusion
Let's summarize - what are these quantities needed for, what are the advantages of such functions? Of course, first of all, they simplify the life of mathematicians and programmers when solving examples, since they allow them to minimize calculations, shorten algorithms, systematize data, and much more.
Where else can this knowledge be useful? In any working specialty: medicine, pharmacology, dentistry, construction, technology, engineering, design, etc.
Construction in negative degree– one of the basic elements of mathematics, which is often encountered when solving algebraic problems. Below are detailed instructions.
How to raise to a negative power - theory
When we raise a number to an ordinary power, we multiply its value several times. For example, 3 3 = 3×3×3 = 27. C negative fraction it's the other way around. General form according to the formula will have next view: a -n = 1/a n . Thus, to raise a number to a negative power, you need to divide one by given number, but to a positive degree.
How to raise to a negative power - examples on ordinary numbers
Keeping the above rule in mind, let's solve a few examples.
4 -2 = 1/4 2 = 1/16
Answer: 4 -2 = 1/16
4 -2 = 1/-4 2 = 1/16.
Answer -4 -2 = 1/16.
But why are the answers in the first and second examples the same? The fact is that when a negative number is raised to an even power (2, 4, 6, etc.), the sign becomes positive. If the degree were even, then the minus would remain:
4 -3 = 1/(-4) 3 = 1/(-64)
How to raise numbers from 0 to 1 to a negative power
Recall that when raising a number in the range from 0 to 1 in positive degree, the value decreases as the degree increases. So for example, 0.5 2 = 0.25. 0.25< 0,5. В случае с отрицательной степенью все обстоит наоборот. При возведении десятичного (дробного) числа в отрицательную степень, значение увеличивается.
Example 3: Calculate 0.5 -2
Solution: 0.5 -2 = 1/1/2 -2 = 1/1/4 = 1×4/1 = 4.
Answer: 0.5 -2 = 4
Analysis (sequence of actions):
- We translate decimal 0.5 to fractional 1/2. It's easier that way.
Raise 1/2 to a negative power. 1/(2) -2 . Divide 1 by 1/(2) 2, we get 1/(1/2) 2 => 1/1/4 = 4
Example 4: Calculate 0.5 -3
Solution: 0.5 -3 = (1/2) -3 = 1/(1/2) 3 = 1/(1/8) = 8
Example 5: Calculate -0.5 -3
Solution: -0.5 -3 = (-1/2) -3 = 1/(-1/2) 3 = 1/(-1/8) = -8
Answer: -0.5 -3 = -8
Based on the 4th and 5th examples, we can draw several conclusions:
- For a positive number in the range from 0 to 1 (example 4), raised to a negative power, whether the power is even or odd is not important, the value of the expression will be positive. At the same time, than more degree, the greater the value.
- For a negative number in the range from 0 to 1 (example 5), raised to a negative power, whether the power is even or odd is not important, the value of the expression will be negative. In this case, the higher the degree, the lower the value.
How to raise to a negative power - a power in the form of a fractional number
Expressions of this type have the following form: a -m/n, where a – regular number, m is the numerator of the degree, n is the denominator of the degree.
Let's look at an example:
Calculate: 8 -1/3
Solution (sequence of actions):
- Let's remember the rule for raising a number to a negative power. We get: 8 -1/3 = 1/(8) 1/3.
- Notice that the denominator has the number 8 in a fractional power. The general form of calculating a fractional power is as follows: a m/n = n √8 m.
- Thus, 1/(8) 1/3 = 1/(3 √8 1). We get cube root out of eight, which is equal to 2. From here, 1/(8) 1/3 = 1/(1/2) = 2.
- Answer: 8 -1/3 = 2
