Erection to negative degree- one of the basic elements of mathematics, which is often found in solving algebraic problems. Below is a detailed instruction.
How to raise to a negative power - theory
When we take a number to the usual power, we multiply its value several times. For example, 3 3 \u003d 3 × 3 × 3 \u003d 27. With a negative fraction, the opposite is true. The general form according to the formula will be as follows: a -n = 1/a n . Thus, to raise a number to a negative power, you need to divide one by the given number, but already to a positive power.
How to raise to a negative power - examples on ordinary numbers
With 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.
The answer is -4 -2 = 1/16.
But why is the answer 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 is preserved:
4 -3 = 1/(-4) 3 = 1/(-64)
How to raise to a negative power - numbers from 0 to 1
Recall that when a number between 0 and 1 is raised to a positive power, the value decreases as the power 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
Parsing (sequence of actions):
- Convert decimal 0.5 to fractional 1/2. It's easier.
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 will draw several conclusions:
- For a positive number between 0 and 1 (example 4) raised to a negative power, whether the power is even or odd, the value of the expression will be positive. In this case, the greater the degree, the greater the value.
- For a negative number between 0 and 1 (Example 5), raised to a negative power, whether the power is even or odd, 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 - the power as a fractional number
Expressions of this type have the following form: a -m/n, where a is an ordinary number, m is the numerator of the degree, n is the denominator of the degree.
Consider an example:
Calculate: 8 -1/3
Solution (sequence of actions):
- Remember the rule for raising a number to a negative power. We get: 8 -1/3 = 1/(8) 1/3 .
- Note that the denominator is 8 to a fractional power. The general form of calculating a fractional degree is as follows: a m/n = n √8 m .
- Thus, 1/(8) 1/3 = 1/(3 √8 1). We get the cube root of eight, which is 2. Based on this, 1/(8) 1/3 = 1/(1/2) = 2.
- Answer: 8 -1/3 = 2
Let's say Achilles runs ten times faster than the tortoise and is a thousand paces behind it. During the time during which Achilles runs this distance, the tortoise crawls a hundred steps in the same direction. When Achilles has run a hundred steps, the tortoise will crawl another ten steps, and so on. The process will continue indefinitely, Achilles will never catch up with the tortoise.
This reasoning became a logical shock for all subsequent generations. Aristotle, Diogenes, Kant, Hegel, Gilbert... All of them, in one way or another, considered Zeno's aporias. The shock was so strong that " ... discussions continue at the present time, the scientific community has not yet managed to come to a common opinion about the essence of paradoxes ... mathematical analysis, set theory, new physical and philosophical approaches were involved in the study of the issue; none of them became a universally accepted solution to the problem ..."[Wikipedia," Zeno's Aporias "]. Everyone understands that they are being fooled, but no one understands what the deception is.
From the point of view of mathematics, Zeno in his aporia clearly demonstrated the transition from the value to. This transition implies applying instead of constants. As far as I understand, the mathematical apparatus for applying variable units of measurement has either not yet been developed, or it has not been applied to Zeno's aporia. The application of our usual logic leads us into a trap. We, by the inertia of thinking, apply constant units of time to the reciprocal. From a physical point of view, it looks like time slowing down to a complete stop at the moment when Achilles catches up with the tortoise. If time stops, Achilles can no longer overtake the tortoise.
If we turn the logic we are used to, everything falls into place. Achilles runs at a constant speed. Each subsequent segment of its path is ten times shorter than the previous one. Accordingly, the time spent on overcoming it is ten times less than the previous one. If we apply the concept of "infinity" in this situation, then it would be correct to say "Achilles will infinitely quickly overtake the tortoise."
How to avoid this logical trap? Remain in constant units of time and do not switch to reciprocal values. In Zeno's language, it looks like this:
In the time it takes Achilles to run a thousand steps, the tortoise crawls a hundred steps in the same direction. For the next time interval, equal to the first, Achilles will run another thousand steps, and the tortoise will crawl a hundred steps. Now Achilles is eight hundred paces ahead of the tortoise.
This approach adequately describes reality without any logical paradoxes. But it is not complete solution Problems. Einstein's statement about the insurmountability of the speed of light is very similar to Zeno's aporia "Achilles and the tortoise". We have yet to study, rethink and solve this problem. And the solution must be sought not in infinitely large numbers, but in units of measurement.
Another interesting aporia of Zeno tells of a flying arrow:
A flying arrow is motionless, since at each moment of time it is at rest, and since it is at rest at every moment of time, it is always at rest.
In this aporia, the logical paradox is overcome very simply - it is enough to clarify that at each moment of time the flying arrow is at rest at different points in space, which, in fact, is movement. There is another point to be noted here. From one photograph of a car on the road, it is impossible to determine either the fact of its movement or the distance to it. To determine the fact of the movement of the car, two photographs taken from the same point at different points in time are needed, but they cannot be used to determine the distance. To determine the distance to the car, you need two photographs taken from different points in space at the same time, but you cannot determine the fact of movement from them (of course, you still need additional data for calculations, trigonometry will help you). What I want to point out in particular is that two points in time and two points in space are two different things that should not be confused as they provide different opportunities for exploration.
Wednesday, July 4, 2018
Very well the differences between set and multiset are described in Wikipedia. We look.
As you can see, "the set cannot have two identical elements", but if there are identical elements in the set, such a set is called a "multiset". Reasonable beings will never understand such logic of absurdity. This is the level of talking parrots and trained monkeys, in which the mind is absent from the word "completely." Mathematicians act as ordinary trainers, preaching their absurd ideas to us.
Once upon a time, the engineers who built the bridge were in a boat under the bridge during the tests of the bridge. If the bridge collapsed, the mediocre engineer died under the rubble of his creation. If the bridge could withstand the load, the talented engineer built other bridges.
No matter how mathematicians hide behind the phrase "mind me, I'm in the house", or rather "mathematics studies abstract concepts", there is one umbilical cord that is inextricably linked to reality. This umbilical cord is money. Applicable mathematical theory sets to the mathematicians themselves.
We studied mathematics very well and now we are sitting at the cash desk, paying salaries. Here a mathematician comes to us for his money. We count the whole amount to him and lay it out on our table into different piles, in which we put bills of the same denomination. Then we take one bill from each pile and give the mathematician his "mathematical salary set". We explain the mathematics that he will receive the rest of the bills only when he proves that the set without identical elements is not equal to the set with identical elements. This is where the fun begins.
First of all, the deputies' logic will work: "you can apply it to others, but not to me!" Further, assurances will begin that there are different banknote numbers on banknotes of the same denomination, which means that they cannot be considered identical elements. Well, we count the salary in coins - there are no numbers on the coins. Here the mathematician will begin to convulsively recall physics: on different coins there is different amount dirt, crystal structure and atomic arrangement of each coin is unique...
And now I have the most interesting question: where is the boundary beyond which elements of a multiset turn into elements of a set and vice versa? Such a line does not exist - everything is decided by shamans, science here is not even close.
Look here. We select football stadiums with the same field area. The area of the fields is the same, which means we have a multiset. But if we consider the names of the same stadiums, we get a lot, because the names are different. As you can see, the same set of elements is both a set and a multiset at the same time. How right? And here the mathematician-shaman-shuller takes out a trump ace from his sleeve and begins to tell us either about a set or a multiset. In any case, he will convince us that he is right.
To understand how modern shamans operate with set theory, tying it to reality, it is enough to answer one question: how do the elements of one set differ from the elements of another set? I will show you, without any "conceivable as not a single whole" or "not conceivable as a single whole."
Sunday, March 18, 2018
The sum of the digits of a number is a dance of shamans with a tambourine, which has nothing to do with mathematics. Yes, in mathematics lessons we are taught to find the sum of the digits of a number and use it, but they are shamans for that, to teach their descendants their skills and wisdom, otherwise shamans will simply die out.
Do you need proof? Open Wikipedia and try to find the "Sum of Digits of a Number" page. She doesn't exist. There is no formula in mathematics by which you can find the sum of the digits of any number. After all, numbers are graphic symbols with which we write numbers, and in the language of mathematics, the task sounds like this: "Find the sum of graphic symbols representing any number." Mathematicians cannot solve this problem, but shamans can do it elementarily.
Let's figure out what and how we do in order to find the sum of digits given number. And so, let's say we have the number 12345. What needs to be done in order to find the sum of the digits of this number? Let's consider all the steps in order.
1. Write down the number on a piece of paper. What have we done? We have converted the number to a number graphic symbol. This is not a mathematical operation.
2. We cut one received picture into several pictures containing separate numbers. Cutting a picture is not a mathematical operation.
3. Convert individual graphic characters to numbers. This is not a mathematical operation.
4. Add up the resulting numbers. Now that's mathematics.
The sum of the digits of the number 12345 is 15. These are the "cutting and sewing courses" from shamans used by mathematicians. But that's not all.
From the point of view of mathematics, it does not matter in which number system we write the number. So, in different systems reckoning, the sum of the digits of the same number will be different. In mathematics, the number system is indicated as a subscript to the right of the number. With a large number of 12345, I don’t want to fool my head, consider the number 26 from the article about. Let's write this number in binary, octal, decimal and hexadecimal number systems. We will not consider each step under a microscope, we have already done that. Let's look at the result.
As you can see, in different number systems, the sum of the digits of the same number is different. This result has nothing to do with mathematics. It's like finding the area of a rectangle in meters and centimeters would give you completely different results.
Zero in all number systems looks the same and has no sum of digits. This is another argument in favor of the fact that . A question for mathematicians: how is it denoted in mathematics that which is not a number? What, for mathematicians, nothing but numbers exists? For shamans, I can allow this, but for scientists, no. Reality is not just about numbers.
The result obtained should be considered as proof that number systems are units of measurement of numbers. After all, we cannot compare numbers with different units of measurement. If the same actions with different units of measurement of the same quantity lead to different results after comparing them, then this has nothing to do with mathematics.
What is real mathematics? This is when the result of a mathematical action does not depend on the value of the number, the unit of measure used, and on who performs this action.
Ouch! Isn't this the women's restroom?
- Young woman! This is a laboratory for studying the indefinite holiness of souls upon ascension to heaven! Nimbus on top and arrow up. What other toilet?
Female... A halo on top and an arrow down is male.
If you have such a work of design art flashing before your eyes several times a day,
Then it is not surprising that you suddenly find a strange icon in your car:
Personally, I make an effort on myself to see minus four degrees in a pooping person (one picture) (composition of several pictures: minus sign, number four, degrees designation). And I don't think that girl is stupid, no who knows physics. She just has an arc stereotype of perception of graphic images. And mathematicians teach us this all the time. Here is an example.
1A is not "minus four degrees" or "one a". This is "pooping man" or the number "twenty-six" in the hexadecimal number system. Those people who constantly work in this number system automatically perceive the number and letter as one graphic symbol.
In continuation of the conversation about the degree of a number, it is logical to deal with finding the value of the degree. This process has been named exponentiation. In this article, we will just study how exponentiation is performed, while touching on all possible exponents - natural, integer, rational and irrational. And by tradition, we will consider in detail the solutions to examples of raising numbers to various degrees.
Page navigation.
What does "exponentiation" mean?
Let's start by explaining what is called exponentiation. Here is the relevant definition.
Definition.
Exponentiation is to find the value of the power of a number.
Thus, finding the value of the power of a with the exponent r and raising the number a to the power of r is the same thing. For example, if the task is “calculate the value of the power (0.5) 5”, then it can be reformulated as follows: “Raise the number 0.5 to the power of 5”.
Now you can go directly to the rules by which exponentiation is performed.
Raising a number to a natural power
In practice, equality based on is usually applied in the form . That is, when raising the number a to fractional degree m/n first, the nth root of the number a is extracted, after which the result is raised to the integer power m.
Consider solutions to examples of raising to a fractional power.
Example.
Calculate the value of the degree.
Solution.
We show two solutions.
First way. By definition of degree with a fractional exponent. We calculate the value of the degree under the sign of the root, after which we extract the cube root: 
.
The second way. By definition of a degree with a fractional exponent and on the basis of the properties of the roots, the equalities are true 
. Now extract the root 
Finally, we raise to an integer power 
.
Obviously, the obtained results of raising to a fractional power coincide.
Answer:
Note that a fractional exponent can be written as a decimal or mixed number, in these cases it should be replaced by the corresponding ordinary fraction, after which exponentiation should be performed.
Example.
Calculate (44.89) 2.5 .
Solution.
We write the exponent in the form common fraction(if necessary, see the article): 
. Now we perform raising to a fractional power: 
Answer:
(44,89) 2,5 =13 501,25107 .
It should also be said that raising numbers to rational powers is a rather laborious process (especially when the numerator and denominator of the fractional exponent contain enough big numbers), which is usually carried out using computer technology.
In conclusion of this paragraph, we will dwell on the construction of the number zero to a fractional power. We gave the following meaning to the fractional degree of zero of the form: for we have 
, while zero to the power m/n is not defined. So zero to a positive fractional power zero, for example, 
. And zero in a fractional negative power does not make sense, for example, the expressions and 0 -4.3 do not make sense.
Raising to an irrational power
Sometimes it becomes necessary to find out the value of the degree of a number with an irrational exponent. In this case, for practical purposes, it is usually sufficient to obtain the value of the degree up to a certain sign. We note right away that in practice this value is calculated using electronic computing technology, since manual raising to an irrational power requires a large number of cumbersome calculations. But nevertheless we will describe in general terms the essence of the actions.
To get an approximate value of the power of a with an irrational exponent, some decimal approximation of the exponent is taken, and the value of the exponent is calculated. This value is the approximate value of the degree of the number a with an irrational exponent. The more accurate the decimal approximation of the number is taken initially, the more accurate the degree value will be in the end.
As an example, let's calculate the approximate value of the power of 2 1.174367... . Let's take the following decimal approximation of an irrational indicator: . Now we raise 2 to a rational power of 1.17 (we described the essence of this process in the previous paragraph), we get 2 1.17 ≈ 2.250116. In this way, 2 1,174367... ≈2 1,17 ≈2,250116 . If we take a more accurate decimal approximation of an irrational exponent, for example, , then we get a more accurate value of the original degree: 2 1,174367... ≈2 1,1743 ≈2,256833 .
Bibliography.
- Vilenkin N.Ya., Zhokhov V.I., Chesnokov A.S., Shvartsburd S.I. Mathematics Zh textbook for 5 cells. educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 7 cells. educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
- Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: a textbook for 9 cells. educational institutions.
- Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
- Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).
One of the main characteristics in algebra, and indeed in all mathematics, is a degree. Of course, in the 21st century, all calculations can be carried out on an online calculator, but it is better to learn how to do it yourself for the development of brains.
In this article, we will consider the most important issues regarding this definition. Namely, we will understand what it is in general and what are its main functions, what properties exist in mathematics.
Let's look at examples of what the calculation looks like, what are the basic formulas. We will analyze 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 a zero degree, irrational, negative, etc.
Online exponentiation calculator
What is the degree of a number
What is meant by the expression "raise a number to a power"?
The degree n of a number a 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 step. = 2 * 2 * 2 = 8;
- 4 2 = 4 in step. two = 4 * 4 = 16;
- 5 4 = 5 in step. four = 5 * 5 * 5 * 5 = 625;
- 10 5 \u003d 10 in 5 step. = 10 * 10 * 10 * 10 * 10 = 100000;
- 10 4 \u003d 10 in 4 step. = 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 grade | 3rd grade |
| 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 |
Degree properties
What is characteristic of such a 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 how to be with addition and subtraction? Everything is simple. First exponentiation is performed, and only then addition and subtraction.
Let's look at examples:
- 3 3 + 2 4 = 27 + 16 = 43;
- 5 2 - 3 2 = 25 - 9 = 16
But in this case, you must first calculate the addition, since there are actions in brackets: (5 + 3) 3 = 8 3 = 512.
How to produce calculations in more complex cases? The order is the same:
- if there are brackets, you need to start with them;
- then exponentiation;
- then perform operations of multiplication, division;
- after addition, subtraction.
There are specific properties that are not characteristic of all degrees:
- The root of the nth degree from the number a to the degree m 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 raising the product of different numbers to a power, the expression will correspond to the product of these numbers to a 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 step, but with a “+” sign.
- If the denominator of a fraction is in a negative power, then this expression will be equal to the product of the numerator and the denominator in a positive power.
- Any number to the power of 0 = 1, and to the step. 1 = to himself.
These rules are important in individual cases, we will consider them in more detail below.
Degree with a negative exponent
What to do with a negative degree, that is, when the indicator is negative?
Based on properties 4 and 5(see point above) it turns out:
A (- n) \u003d 1 / A n, 5 (-2) \u003d 1/5 2 \u003d 1/25.
And vice versa:
1 / A (- n) \u003d A n, 1 / 2 (-3) \u003d 2 3 \u003d 8.
What if it's a fraction?
(A / B) (- n) = (B / A) n , (3 / 5) (-2) = (5 / 3) 2 = 25 / 9.
Degree with a 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.
Also, 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 all the specific features described above, are also characteristic of them.
Fractional degree
This view can be written as a scheme: A m / n. It is read as: the root of the nth degree of the number A to the power of m.
With a fractional indicator, you can do anything: reduce, decompose into parts, raise to another degree, etc.
Degree with irrational exponent
Let α be an irrational number and А ˃ 0.
To understand the essence of the degree with such an indicator, Let's look at different possible cases:
- A \u003d 1. The result will be equal to 1. Since there is an axiom - 1 is equal to one in all powers;
А r 1 ˂ А α ˂ А r 2 , r 1 ˂ r 2 are rational numbers;
- 0˂А˂1.
In this case, vice versa: А r 2 ˂ А α ˂ А r 1 under the same conditions as in the second paragraph.
For example, the exponent is the number π. It is rational.
r 1 - in this case it is equal to 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 the mathematical operations and specific properties described above.
Conclusion
Let's summarize - what are these values for, what are the advantages of such functions? Of course, first of all, they simplify the lives of mathematicians and programmers when solving examples, since they allow minimizing calculations, reducing algorithms, systematizing data, and much more.
Where else can this knowledge be useful? In any working specialty: medicine, pharmacology, dentistry, construction, technology, engineering, design, etc.
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 in a 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 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 Chess board 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. Unexpectedly, right?) Draw a pool: a bottom one meter in size and a meter deep and try to calculate how many cubes meter by meter in total 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 the 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 and in general view 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 a number with a natural exponent
You probably already guessed it: because the exponent is 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 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 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, endless decimal. 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).
- Any number to the first power is equal to itself:
- To square a number is to multiply it by itself:
- 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 does not 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 circle 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 “preparation of a number”, namely a 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, in science, a degree with a complex exponent is often used, 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:
In 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, apply regular properties 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. It is possible to formulate such simple rules:
- even degree, - number positive.
- Negative number raised to odd degree, - number negative.
- A positive number to any power 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: it does not 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 a negative integer - it's 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, in science, a degree with a complex exponent is often used, 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:
- Remember the difference of squares formula. Answer: .
- We bring fractions to the same form: either both decimals, or both ordinary ones. We get, for example: .
- 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!
