Sections: Mathematics
Lesson type: lesson of generalization and systematization of knowledge
Goals:
Equipment: computer, multimedia projector, interactive whiteboard, presentation of “Degrees” for mental calculation, task cards, handouts.
Lesson plan:
Lesson progress
I. Organizational moment
Communicate the topic and objectives of the lesson.
In previous lessons you discovered amazing world degrees, learned to multiply and divide degrees, and raise them to a power. Today we must consolidate the acquired knowledge by solving examples.
II. Repetition of rules(orally)
- Give the definition of degree with a natural exponent? (Power of number A with a natural exponent greater than 1 is called a product n factors, each of which is equal A.)
- How to multiply two powers? (To multiply powers with the same bases, you must leave the base the same and add the exponents.)
- How to divide degree by degree? (To divide powers with the same bases, you need to leave the base the same and subtract the exponents.)
- How to raise a product to a power? (To raise a product to a power, you need to raise each factor to that power)
- How to raise a degree to a power? (To raise a power to a power, you need to leave the base the same and multiply the exponents)
- for the eyes
- for the neck
- for hands
- for the torso
- for feet
III. Oral counting(by multimedia)
IV. Historical background
All problems are from the Ahmes papyrus, which was written around 1650 BC. e. related to construction practice, demarcation of land plots, etc. Tasks are grouped by topic. These are mainly problems of finding the areas of a triangle, quadrilaterals and a circle, various operations with integers and fractions, proportional division, finding ratios, there is also raising to different powers, solving equations of the first and second degree with one unknown.
There is a complete lack of any explanation or evidence. The desired result is either given directly or a short algorithm for calculating it is given. This method of presentation, typical of science in the countries of the ancient East, suggests that mathematics there developed through generalizations and guesses that did not form any general theory. However, the papyrus contains a whole series evidence that Egyptian mathematicians knew how to extract roots and raise to powers, solve equations, and even knew the rudiments of algebra.
V. Work at the board
Find the meaning of the expression in a rational way:
Calculate the value of the expression:
VI. Physical education minute
VII. Problem solving(with display on the interactive whiteboard)
Is the root of the equation a positive number?
xn--i1abbnckbmcl9fb.xn--p1ai
Formulas of powers and roots.
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. By multiplying degrees with the same base, their indicators are added:
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:
5. Raising a power to a power, the exponents are multiplied:
Each formula above is true in the directions from left to right and vice versa.
Operations with roots.
1. The root of the product of several factors is equal to the product of the roots of these factors:
2. The root of a ratio is equal to the ratio of the dividend and the divisor of the 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:
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 4: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.
The power of any number, not equal to zero, with a zero exponent equals one.
To build real number A to the degree m/n, you need to extract the root n-th degree from m-th power of this number A:
Degree formulas.
6. a — n = - division of degrees;
7. - division of degrees;
8. a 1/n = ;
Degrees of the rule of action with degrees
1. The degree of the product of two or more factors is equal to the product of the degrees of these factors (with the same exponent):
(abc…) n = a n b n c n …
Example 1. (7 2 10) 2 = 7 2 2 2 10 2 = 49 4 100 = 19600. Example 2. (x 2 –a 2) 3 = [(x +a)(x – a)] 3 =( x +a) 3 (x - a) 3
In practice, the reverse conversion is more important:
a n b n c n … = (abc…) n
those. the product of identical powers of several quantities is equal to the same power of the product of these quantities.
Example 3. 
Example 4. (a +b) 2 (a 2 – ab +b 2) 2 =[(a +b)(a 2 – ab +b 2)] 2 =(a 3 +b 3) 2
2. The power of a quotient (fraction) is equal to the quotient of dividing the same power of the divisor by the same power:
Example 5. 
Example 6. 
Reverse conversion:. Example 7. 
. Example 8. 
.
3. When multiplying degrees with the same bases, the exponents of the degrees are added:
Example 9.2 2 2 5 =2 2+5 =2 7 =128. Example 10. (a – 4c +x) 2 (a – 4c +x) 3 =(a – 4c + x) 5.
4. When dividing powers with the same bases, the exponent of the divisor is subtracted from the exponent of the dividend
Example 11. 12 5:12 3 =12 5-3 =12 2 =144. Example 12. (x-y) 3:(x-y) 2 =x-y.
5. When raising a degree to a power, the exponents are multiplied:
Example 13. (2 3) 2 =2 6 =64. Example 14. 
www.maths.yfa1.ru
Powers and roots
Operations with powers and roots. Degree with negative ,
zero and fractional indicator. About expressions that have no meaning.
Operations with degrees.
1. When multiplying powers with the same base, their exponents are added:
a m · a n = a m + n .
2. When dividing degrees with the same base, their exponents are deducted .
3. The degree of the product of two or more factors is equal to the product of the degrees of these factors.
4. The degree of a ratio (fraction) is equal to the ratio of the degrees of the dividend (numerator) and divisor (denominator):
(a/b) n = a n / b n .
5. When raising a power to a power, their exponents are multiplied:
All the above formulas are read and executed in both directions from left to right and vice versa.
EXAMPLE (2 3 5 / 15)² = 2² · 3² · 5² / 15² = 900 / 225 = 4 .
Operations with roots. In all the formulas below, the symbol means arithmetic root (the radical expression is positive).
1. The root of the product of several factors is equal to the product of the roots of these factors:
2. The root of a ratio is equal to the ratio of the roots of the dividend and the divisor:
3. When raising a root to a power, it is enough to raise to this power radical number:
4. If you increase the degree of the root by m times and at the same time raise the radical number to the mth power, then the value of the root will not change:
5. If you reduce the degree of the root by m times and simultaneously extract the mth root of the radical number, then the value of the root will not change:
Expanding the concept of degree. So far we have considered degrees only with natural exponents; but operations with powers and roots can also lead to negative, zero And fractional indicators. All these exponents require additional definition.
A degree with a negative exponent. The power of a certain number with a negative (integer) exponent is defined as one divided by the power of the same number with an exponent equal to the absolute value of the negative exponent:
Now the formula a m : a n = a m - n can be used not only for m, more than n, but also with m, less than n .
EXAMPLE a 4: a 7 = a 4 — 7 = a — 3 .
If we want the formula a m : a n = a m — n was fair when m = n, we need a definition of degree zero.
A degree with a zero index. The power of any non-zero number with exponent zero is 1.
EXAMPLES. 2 0 = 1, ( – 5) 0 = 1, (– 3 / 5) 0 = 1.
Degree with a fractional exponent. In order to raise a real number a to the power m / n, you need to extract the nth root of the mth power of this number a:
About expressions that have no meaning. There are several such expressions.
Where a ≠ 0 , does not exist.
In fact, if we assume that x is a certain number, then in accordance with the definition of the division operation we have: a = 0· x, i.e. a= 0, which contradicts the condition: a ≠ 0
— any number.
In fact, if we assume that this expression is equal to some number x, then according to the definition of the division operation we have: 0 = 0 · x. But this equality occurs when any number x, which was what needed to be proven.
0 0 — any number.
Solution. Let's consider three main cases:
1) x = 0 – this value does not satisfy this equation
2) when x> 0 we get: x/x= 1, i.e. 1 = 1, which means
What x– any number; but taking into account that in
in our case x> 0, the answer is x > 0 ;
Properties of degree
We remind you that in this lesson we will understand properties of degrees with natural indicators and zero. Powers with rational exponents and their properties will be discussed in lessons for 8th grade.
A power with a natural exponent has several important properties that allow us to simplify calculations in examples with powers.
Property No. 1
Product of powers
When multiplying powers with the same bases, the base remains unchanged, and the exponents of the powers are added.
a m · a n = a m + n, where “a” is any number, and “m”, “n” are any natural numbers.
This property of powers also applies to the product of three or more powers.
- Simplify the expression.
b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15 - Present it as a degree.
6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17 - Present it as a degree.
(0.8) 3 · (0.8) 12 = (0.8) 3 + 12 = (0.8) 15 - Write the quotient as a power
(2b) 5: (2b) 3 = (2b) 5 − 3 = (2b) 2 - Calculate.
Please note that in the specified property we were talking only about the multiplication of powers with the same bases. It does not apply to their addition.
You cannot replace the sum (3 3 + 3 2) with 3 5. This is understandable if
count (3 3 + 3 2) = (27 + 9) = 36, and 3 5 = 243
Property No. 2
Partial degrees
When dividing powers with the same base, the base remains unchanged, and the exponent of the divisor is subtracted from the exponent of the dividend.
11 3 − 2 4 2 − 1 = 11 4 = 44
Example. Solve the equation. We use the property of quotient powers.
3 8: t = 3 4
Answer: t = 3 4 = 81
Using properties No. 1 and No. 2, you can easily simplify expressions and perform calculations.
Example. Simplify the expression.
4 5m + 6 4 m + 2: 4 4m + 3 = 4 5m + 6 + m + 2: 4 4m + 3 = 4 6m + 8 − 4m − 3 = 4 2m + 5
Example. Find the value of an expression using the properties of exponents.
2 11 − 5 = 2 6 = 64
Please note that in Property 2 we were only talking about dividing powers with the same bases.
You cannot replace the difference (4 3 −4 2) with 4 1. This is understandable if you calculate (4 3 −4 2) = (64 − 16) = 48, and 4 1 = 4
Property No. 3
Raising a degree to a power
When raising a degree to a power, the base of the degree remains unchanged, and the exponents are multiplied.
(a n) m = a n · m, where “a” is any number, and “m”, “n” are any natural numbers.
(a 4) 6 = a 4 6 = a 24
By the property of raising a degree to a power It is known that when raised to a power, exponents are multiplied, which means:
Properties 4
Product power
When a power is raised to a product power, each factor is raised to that power and the results are multiplied.
(a b) n = a n b n, where “a”, “b” are any rational numbers; "n" is any natural number.
- Example 1.
(6 a 2 b 3 c) 2 = 6 2 a 2 2 b 3 2 c 1 2 = 36 a 4 b 6 c 2 - Example 2.
(−x 2 y) 6 = ((−1) 6 x 2 6 y 1 6) = x 12 y 6 - Example. Calculate.
2 4 5 4 = (2 5) 4 = 10 4 = 10,000 - Example. Calculate.
0.5 16 2 16 = (0.5 2) 16 = 1 - Example. Present the expression as a quotient of powers.
(5: 3) 12 = 5 12: 3 12
Please note that property No. 4, like other properties of degrees, is also applied in reverse order.
(a n · b n)= (a · b) n
That is, to multiply powers with the same exponents, you can multiply the bases, but leave the exponent unchanged.
In more complex examples, there may be cases where multiplication and division must be performed over powers with for different reasons and various indicators. In this case, we advise you to do the following.
For example, 4 5 3 2 = 4 3 4 2 3 2 = 4 3 (4 3) 2 = 64 12 2 = 64 144 = 9216
An example of raising a decimal to a power.
4 21 (−0.25) 20 = 4 4 20 (−0.25) 20 = 4 (4 (−0.25)) 20 = 4 (−1) 20 = 4 1 = 4
Properties 5
Power of a quotient (fraction)
To raise a quotient to a power, you can raise the dividend and the divisor separately to this power, and divide the first result by the second.
(a: b) n = a n: b n, where “a”, “b” are any rational numbers, b ≠ 0, n - any natural number.
We remind you that a quotient can be represented as a fraction. Therefore, we will dwell on the topic of raising a fraction to a power in more detail on the next page.
Solving exponential equations. Examples.
Attention!
There are additional
materials in Special Section 555.
For those who are very "not very..."
And for those who “very much…”)
What's happened exponential equation? This is an equation in which the unknowns (x's) and expressions with them are in indicators some degrees. And only there! This is important.
Here you go examples of exponential equations:
3 x 2 x = 8 x+3
Pay attention! In the bases of degrees (below) - only numbers. IN indicators degrees (above) - a wide variety of expressions with an X. If, suddenly, an X appears in the equation somewhere other than an indicator, for example:
this will be an equation mixed type. Such equations do not have clear rules for solving them. We will not consider them for now. Here we will deal with solving exponential equations in its purest form.
In fact, even pure exponential equations are not always solved clearly. But there are certain types of exponential equations that can and should be solved. These are the types we will consider.
Solving simple exponential equations.
First, let's solve something very basic. For example:
Even without any theories, by simple selection it is clear that x = 2. Nothing more, right!? No other value of X works. Now let's look at the solution to this tricky exponential equation:
What have we done? We, in fact, simply threw out the same bases (triples). Completely thrown out. And, the good news is, we hit the nail on the head!
Indeed, if in an exponential equation there are left and right identical numbers in any powers, these numbers can be removed and the exponents can be equalized. Mathematics allows. It remains to solve a much simpler equation. Great, right?)
However, let us remember firmly: You can remove bases only when the base numbers on the left and right are in splendid isolation! Without any neighbors and coefficients. Let's say in the equations:
2 x +2 x+1 = 2 3, or
twos cannot be removed!
Well, we have mastered the most important thing. How to move on from evil demonstrative expressions to simpler equations.
"Those are the times!" - you say. “Who would give such a primitive lesson on tests and exams!?”
I have to agree. Nobody will. But now you know where to aim when solving tricky examples. It is necessary to bring it to the form where the same base number is on the left and on the right. Then everything will be easier. Actually, this is a classic of mathematics. We take the original example and transform it to the desired one us mind. According to the rules of mathematics, of course.
Let's look at examples that require some additional effort to reduce them to the simplest. Let's call them simple exponential equations.
Solving simple exponential equations. Examples.
When solving exponential equations, the main rules are actions with degrees. Without knowledge of these actions nothing will work.
To actions with degrees, one must add personal observation and ingenuity. Do we need the same base numbers? So we look for them in the example in explicit or encrypted form.
Let's see how this is done in practice?
Let us be given an example:
2 2x - 8 x+1 = 0
The first keen look is at grounds. They... They are different! Two and eight. But it’s too early to become discouraged. It's time to remember that
Two and eight are relatives in degree.) It is quite possible to write:
8 x+1 = (2 3) x+1
If we recall the formula from operations with degrees:
(a n) m = a nm ,
this works out great:
8 x+1 = (2 3) x+1 = 2 3(x+1)
The original example began to look like this:
2 2x - 2 3(x+1) = 0
We transfer 2 3 (x+1) to the right (no one has canceled the elementary operations of mathematics!), we get:
2 2x = 2 3(x+1)
That's practically all. Removing the bases:
We solve this monster and get
This is the correct answer.
In this example, knowing the powers of two helped us out. We identified in eight there is an encrypted two. This technique (encryption common grounds under different numbers) is a very popular technique in exponential equations! Yes, and in logarithms too. You must be able to recognize powers of other numbers in numbers. This is extremely important for solving exponential equations.
The fact is that raising any number to any power is not a problem. Multiply, even on paper, and that’s it. For example, anyone can raise 3 to the fifth power. 243 will work out if you know the multiplication table.) But in exponential equations, much more often it is not necessary to raise to a power, but vice versa... Find out what number to what degree is hidden behind the number 243, or, say, 343... No calculator will help you here.
You need to know the powers of some numbers by sight, right... Let's practice?
Determine what powers and what numbers the numbers are:
2; 8; 16; 27; 32; 64; 81; 100; 125; 128; 216; 243; 256; 343; 512; 625; 729, 1024.
Answers (in a mess, of course!):
5 4 ; 2 10 ; 7 3 ; 3 5 ; 2 7 ; 10 2 ; 2 6 ; 3 3 ; 2 3 ; 2 1 ; 3 6 ; 2 9 ; 2 8 ; 6 3 ; 5 3 ; 3 4 ; 2 5 ; 4 4 ; 4 2 ; 2 3 ; 9 3 ; 4 5 ; 8 2 ; 4 3 ; 8 3 .
If you look closely, you can see a strange fact. There are significantly more answers than tasks! Well, it happens... For example, 2 6, 4 3, 8 2 - that's all 64.
Let us assume that you have taken note of the information about familiarity with numbers.) Let me also remind you that to solve exponential equations we use all stock of mathematical knowledge. Including those from junior and middle classes. You didn’t go straight to high school, right?)
For example, when solving exponential equations, putting the common factor out of brackets often helps (hello to 7th grade!). Let's look at an example:
3 2x+4 -11 9 x = 210
And again, the first glance is at the foundations! The bases of the degrees are different... Three and nine. And we want them to be the same. Well, in this case the desire is completely fulfilled!) Because:
9 x = (3 2) x = 3 2x
Using the same rules for dealing with degrees:
3 2x+4 = 3 2x ·3 4
That’s great, you can write it down:
3 2x 3 4 - 11 3 2x = 210
We gave an example for the same reasons. And what next!? You can't throw out threes... Dead end?
Not at all. Remember the most universal and powerful decision rule everyone math assignments:
If you don’t know what you need, do what you can!
Look, everything will work out).
What's in this exponential equation Can do? Yes, on the left side it just begs to be taken out of brackets! The overall multiplier of 3 2x clearly hints at this. Let's try, and then we'll see:
3 2x (3 4 - 11) = 210
3 4 - 11 = 81 - 11 = 70
The example keeps getting better and better!
We remember that to eliminate grounds we need a pure degree, without any coefficients. The number 70 bothers us. So we divide both sides of the equation by 70, we get:
Oops! Everything got better!
This is the final answer.
It happens, however, that taxiing on the same basis is achieved, but their elimination is not possible. This happens in other types of exponential equations. Let's master this type.
Replacing a variable in solving exponential equations. Examples.
Let's solve the equation:
4 x - 3 2 x +2 = 0
First - as usual. Let's move on to one base. To a deuce.
4 x = (2 2) x = 2 2x
We get the equation:
2 2x - 3 2 x +2 = 0
And this is where we hang out. The previous techniques will not work, no matter how you look at it. We'll have to get out of the arsenal another powerful and universal method. It's called variable replacement.
The essence of the method is surprisingly simple. Instead of one complex icon (in our case - 2 x) we write another, simpler one (for example - t). Such a seemingly meaningless replacement leads to amazing results!) Everything just becomes clear and understandable!
So let
Then 2 2x = 2 x2 = (2 x) 2 = t 2
In our equation we replace all powers with x's by t:
Well, is it dawning on you?) Quadratic equations Have you forgotten yet? Solving through the discriminant, we get:
The main thing here is not to stop, as happens... This is not the answer yet, we need an x, not a t. Let's return to the X's, i.e. we make a reverse replacement. First for t 1:
Therefore,
One root was found. We are looking for the second one from t 2:
Hm... 2 x on the left, 1 on the right... Problem? Not at all! It is enough to remember (from operations with powers, yes...) that a unit is any number to the zero power. Any. Whatever is needed, we will install it. We need a two. Means:
That's it now. We got 2 roots:
This is the answer.
At solving exponential equations at the end sometimes you end up with some kind of awkward expression. Type:
From seven to two through simple degree it doesn't work. They are not relatives... How can we be? Someone may be confused... But the person who read on this site the topic “What is a logarithm?” , just smiles sparingly and writes down with a firm hand the absolutely correct answer:
There cannot be such an answer in tasks “B” on the Unified State Examination. There a specific number is required. But in tasks “C” it’s easy.
This lesson provides examples of solving the most common exponential equations. Let's highlight the main points.
1. First of all, we look at grounds degrees. We are wondering if it is possible to make them identical. Let's try to do this by actively using actions with degrees. Don't forget that numbers without x's can also be converted to powers!
2. We try to bring the exponential equation to the form when on the left and on the right there are identical numbers in any powers. We use actions with degrees And factorization. What can be counted in numbers, we count.
3. If the second tip doesn’t work, try using variable replacement. The result may be an equation that can be easily solved. Most often - square. Or fractional, which also reduces to square.
4. To successfully solve exponential equations, you need to know the powers of some numbers by sight.
As usual, at the end of the lesson you are invited to decide a little.) On your own. From simple to complex.
Solve exponential equations:
More difficult:
2 x+3 - 2 x+2 - 2 x = 48
9 x - 8 3 x = 9
2 x - 2 0.5x+1 - 8 = 0
Find the product of roots:
2 3's + 2 x = 9
Did it work?
Well then the most complicated example(decided, however, in the mind...):
7 0.13x + 13 0.7x+1 + 2 0.5x+1 = -3
What's more interesting? Then here's a bad example for you. Quite worthy of increased difficulty. Let me hint that in this example, what saves you is ingenuity and the most universal rule for solving all mathematical problems.)
2 5x-1 3 3x-1 5 2x-1 = 720 x
A simpler example, for relaxation):
9 2 x - 4 3 x = 0
And for dessert. Find the sum of the roots of the equation:
x 3 x - 9x + 7 3 x - 63 = 0
Yes, yes! This is a mixed type equation! Which we did not consider in this lesson. Why consider them, they need to be solved!) This lesson is quite enough to solve the equation. Well, you need ingenuity... And may seventh grade help you (this is a hint!).
Answers (in disarray, separated by semicolons):
1; 2; 3; 4; there are no solutions; 2; -2; -5; 4; 0.
Is everything successful? Great.
Any problems? No question! Special Section 555 solves all these exponential equations with detailed explanations. What, why, and why. And, of course, there is additional valuable information on working with all sorts of exponential equations. Not just these ones.)
One last fun question to consider. In this lesson we worked with exponential equations. Why didn’t I say a word about ODZ here? In equations, this is a very important thing, by the way...
If you like this site...
By the way, I have a couple more interesting sites for you.)
You can practice solving examples and find out your level. Testing with instant verification. Let's learn - with interest!)
You can get acquainted with functions and derivatives.
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 issues related to this definition. Namely, we will 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 construction natural numbers V positive degrees– “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 hold 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.
There are specific properties that are not characteristic of 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 in negative degree, then this expression will be equal to the product of the numerator and the denominator to a positive power.
- Any number to the power of 0 = 1, and to the power of. 1 = to yourself.
These rules are important in in some cases, we will consider them in more detail below.
Degree with a negative exponent
What to do with a 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 negative number 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 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 the 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.
It is obvious that numbers with powers can be added like other quantities , by adding them one after another with their signs.
So, the sum of a 3 and b 2 is a 3 + b 2.
The sum of a 3 - b n and h 5 -d 4 is a 3 - b n + h 5 - d 4.
Odds equal powers of equal variables can be added or subtracted.
So, the sum of 2a 2 and 3a 2 is equal to 5a 2.
It is also obvious that if you take two squares a, or three squares a, or five squares a.
But degrees various variables And various degrees identical variables, must be composed by adding them with their signs.
So, the sum of a 2 and a 3 is the sum of a 2 + a 3.
It is obvious that the square of a, and the cube of a, is not equal to twice the square of a, but to twice the cube of a.
The sum of a 3 b n and 3a 5 b 6 is a 3 b n + 3a 5 b 6.
Subtraction powers are carried out in the same way as addition, except that the signs of the subtrahends must be changed accordingly.
Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 - 4h 2 b 6 = -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6
Multiplying powers
Numbers with powers can be multiplied, like other quantities, by writing them one after the other, with or without a multiplication sign between them.
Thus, the result of multiplying a 3 by b 2 is a 3 b 2 or aaabb.
Or:
x -3 ⋅ a m = a m x -3
3a 6 y 2 ⋅ (-2x) = -6a 6 xy 2
a 2 b 3 y 2 ⋅ a 3 b 2 y = a 2 b 3 y 2 a 3 b 2 y
The result in the last example can be ordered by adding identical variables.
The expression will take the form: a 5 b 5 y 3.
By comparing several numbers (variables) with powers, we can see that if any two of them are multiplied, then the result is a number (variable) with a power equal to amount degrees of terms.
So, a 2 .a 3 = aa.aaa = aaaaa = a 5 .
Here 5 is the power of the result of the multiplication, equal to 2 + 3, the sum of the powers of the terms.
So, a n .a m = a m+n .
For a n , a is taken as a factor as many times as the power of n;
And a m is taken as a factor as many times as the degree m is equal to;
That's why, powers with the same bases can be multiplied by adding the exponents of the powers.
So, a 2 .a 6 = a 2+6 = a 8 . And x 3 .x 2 .x = x 3+2+1 = x 6 .
Or:
4a n ⋅ 2a n = 8a 2n
b 2 y 3 ⋅ b 4 y = b 6 y 4
(b + h - y) n ⋅ (b + h - y) = (b + h - y) n+1
Multiply (x 3 + x 2 y + xy 2 + y 3) ⋅ (x - y).
Answer: x 4 - y 4.
Multiply (x 3 + x - 5) ⋅ (2x 3 + x + 1).
This rule is also true for numbers whose exponents are negative.
1. So, a -2 .a -3 = a -5 . This can be written as (1/aa).(1/aaa) = 1/aaaaa.
2. y -n .y -m = y -n-m .
3. a -n .a m = a m-n .
If a + b are multiplied by a - b, the result will be a 2 - b 2: that is
The result of multiplying the sum or difference of two numbers equal to the sum or the difference of their squares.
If you multiply the sum and difference of two numbers raised to square, the result will be equal to the sum or difference of these numbers in fourth degrees.
So, (a - y).(a + y) = a 2 - y 2.
(a 2 - y 2)⋅(a 2 + y 2) = a 4 - y 4.
(a 4 - y 4)⋅(a 4 + y 4) = a 8 - y 8.
Division of degrees
Numbers with powers can be divided like other numbers, by subtracting from the dividend, or by placing them in fraction form.
Thus, a 3 b 2 divided by b 2 is equal to a 3.
Or:
$\frac(9a^3y^4)(-3a^3) = -3y^4$
$\frac(a^2b + 3a^2)(a^2) = \frac(a^2(b+3))(a^2) = b + 3$
$\frac(d\cdot (a - h + y)^3)((a - h + y)^3) = d$
Writing a 5 divided by a 3 looks like $\frac(a^5)(a^3)$. But this is equal to a 2 . In a series of numbers
a +4 , a +3 , a +2 , a +1 , a 0 , a -1 , a -2 , a -3 , a -4 .
any number can be divided by another, and the exponent will be equal to difference indicators of divisible numbers.
When dividing degrees with the same base, their exponents are subtracted..
So, y 3:y 2 = y 3-2 = y 1. That is, $\frac(yyy)(yy) = y$.
And a n+1:a = a n+1-1 = a n . That is, $\frac(aa^n)(a) = a^n$.
Or:
y 2m: y m = y m
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b +y) n-3
The rule is also true for numbers with negative values of degrees.
The result of dividing a -5 by a -3 is a -2.
Also, $\frac(1)(aaaaa) : \frac(1)(aaa) = \frac(1)(aaaaa).\frac(aaa)(1) = \frac(aaa)(aaaaa) = \frac (1)(aa)$.
h 2:h -1 = h 2+1 = h 3 or $h^2:\frac(1)(h) = h^2.\frac(h)(1) = h^3$
It is necessary to master multiplication and division of powers very well, since such operations are very widely used in algebra.
Examples of solving examples with fractions containing numbers with powers
1. Reduce the exponents by $\frac(5a^4)(3a^2)$ Answer: $\frac(5a^2)(3)$.
2. Decrease the exponents by $\frac(6x^6)(3x^5)$. Answer: $\frac(2x)(1)$ or 2x.
3. Reduce the exponents a 2 /a 3 and a -3 /a -4 and bring to a common denominator.
a 2 .a -4 is a -2 the first numerator.
a 3 .a -3 is a 0 = 1, the second numerator.
a 3 .a -4 is a -1 , the common numerator.
After simplification: a -2 /a -1 and 1/a -1 .
4. Reduce the exponents 2a 4 /5a 3 and 2 /a 4 and bring to a common denominator.
Answer: 2a 3 /5a 7 and 5a 5 /5a 7 or 2a 3 /5a 2 and 5/5a 2.
5. Multiply (a 3 + b)/b 4 by (a - b)/3.
6. Multiply (a 5 + 1)/x 2 by (b 2 - 1)/(x + a).
7. Multiply b 4 /a -2 by h -3 /x and a n /y -3 .
8. Divide a 4 /y 3 by a 3 /y 2 . Answer: a/y.
9. Divide (h 3 - 1)/d 4 by (d n + 1)/h.
Lesson on the topic: "Rules of multiplication and division of powers with the same and different exponents. Examples"
Additional materials
Dear users, do not forget to leave your comments, reviews, wishes. All materials have been checked by an anti-virus program.
Educational aids and simulators in the Integral online store for grade 7
Manual for the textbook Yu.N. Makarycheva Manual for the textbook by A.G. Mordkovich
Purpose of the lesson: learn to perform operations with powers of numbers.
First, let's remember the concept of "power of number". An expression of the form $\underbrace( a * a * \ldots * a )_(n)$ can be represented as $a^n$.
The converse is also true: $a^n= \underbrace( a * a * \ldots * a )_(n)$.
This equality is called “recording the degree as a product.” It will help us determine how to multiply and divide powers.
Remember:
a– the basis of the degree.
n– exponent.
If n=1, which means the number A took once and accordingly: $a^n= 1$.
If n= 0, then $a^0= 1$.
We can find out why this happens when we get acquainted with the rules of multiplication and division of powers.
Multiplication rules
a) If powers with the same base are multiplied.To get $a^n * a^m$, we write the degrees as a product: $\underbrace( a * a * \ldots * a )_(n) * \underbrace( a * a * \ldots * a )_(m )$.
The figure shows that the number A took n+m times, then $a^n * a^m = a^(n + m)$.
Example.
$2^3 * 2^2 = 2^5 = 32$.
This property is convenient to use to simplify the work when raising a number to a higher power.
Example.
$2^7= 2^3 * 2^4 = 8 * 16 = 128$.
b) If degrees with different bases, but the same exponent are multiplied.
To get $a^n * b^n$, we write the degrees as a product: $\underbrace( a * a * \ldots * a )_(n) * \underbrace( b * b * \ldots * b )_(m )$.
If we swap the factors and count the resulting pairs, we get: $\underbrace( (a * b) * (a * b) * \ldots * (a * b) )_(n)$.
So $a^n * b^n= (a * b)^n$.
Example.
$3^2 * 2^2 = (3 * 2)^2 = 6^2= 36$.
Division rules
a) The basis of the degree is the same, the indicators are different.Consider dividing a power with a larger exponent by dividing a power with a smaller exponent.
So, we need $\frac(a^n)(a^m)$, Where n>m.
Let's write the degrees as a fraction:
$\frac(\underbrace( a * a * \ldots * a )_(n))(\underbrace( a * a * \ldots * a )_(m))$.
For convenience, we write the division as a simple fraction.Now let's reduce the fraction.
It turns out: $\underbrace( a * a * \ldots * a )_(n-m)= a^(n-m)$.
Means, $\frac(a^n)(a^m)=a^(n-m)$.
This property will help explain the situation with raising a number to the zero power. Let's assume that n=m, then $a^0= a^(n-n)=\frac(a^n)(a^n) =1$.
Examples.
$\frac(3^3)(3^2)=3^(3-2)=3^1=3$.
$\frac(2^2)(2^2)=2^(2-2)=2^0=1$.
b) The bases of the degree are different, the indicators are the same.
Let's say we need $\frac(a^n)( b^n)$. Let's write powers of numbers as fractions:
$\frac(\underbrace( a * a * \ldots * a )_(n))(\underbrace( b * b * \ldots * b )_(n))$.
For convenience, let's imagine.
Using the property of fractions, we divide the large fraction into the product of small ones, we get.
$\underbrace( \frac(a)(b) * \frac(a)(b) * \ldots * \frac(a)(b) )_(n)$.
Accordingly: $\frac(a^n)( b^n)=(\frac(a)(b))^n$.
Example.
$\frac(4^3)( 2^3)= (\frac(4)(2))^3=2^3=8$.
