Degree formulas used in the process of reduction and simplification complex expressions, in solving equations and inequalities.
Number c is n-th power of a number a When:
Operations with degrees.
1. Multiplying powers of c the same basis their indicators add up:
a m·a n = a m + n .
2. When dividing degrees with the same base, their exponents are subtracted:
3. Power of the product of 2 or more factors is equal to the product of the powers of these factors:
(abc…) n = a n · b n · c n …
4. The degree of a fraction is equal to the ratio of the degrees of the dividend and the divisor:
(a/b) n = a n /b n .
5. Raising a power to a power, the exponents are multiplied:
(a m) n = a m n .
Each formula above is true in the directions from left to right and vice versa.
For example. (2 3 5/15)² = 2² 3² 5²/15² = 900/225 = 4.
Operations with roots.
1. The root of the product of several factors is equal to the product of the roots of these factors:
2. 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:
A degree with a negative exponent. The power of a certain number with a non-positive (integer) exponent is defined as one divided by the power of the same number with an exponent equal to the absolute value of the non-positive exponent:
Formula a m:a n =a m - n can be used not only for m> n, but also with m< n.
For example. a4:a 7 = a 4 - 7 = a -3.
To formula a m:a n =a m - n became fair when m=n, the presence of zero degree is required.
A degree with a zero index. The power of any number not equal to zero with a zero exponent is equal to one.
For example. 2 0 = 1,(-5) 0 = 1,(-3/5) 0 = 1.
Degree with a fractional exponent. To raise a real number A to the degree m/n, you need to extract the root n th degree of m-th power of this number A.
A number raised to a power They call a number that is multiplied by itself several times.
Power of a number with a negative value (a - n) can be determined in a similar way to how the power of the same number with a positive exponent is determined (a n) . However, it also requires additional definition. The formula is defined as:
a-n = (1/a n)
The properties of negative powers of numbers are similar to powers with a positive exponent. Presented equation a m/a n= a m-n may be fair as
« Nowhere, as in mathematics, does the clarity and accuracy of the conclusion allow a person to wriggle out of an answer by talking around the question».
A. D. Alexandrov
at n more m , and with m more n . Let's look at an example: 7 2 -7 5 =7 2-5 =7 -3 .
First you need to determine the number that acts as a definition of the degree. b=a(-n) . In this example -n is an exponent b - the desired numerical value, a - the base of the degree in the form of a natural numeric value. Then determine the modulus, that is, the absolute value negative number, which acts as an exponent. Calculate the degree of a given number relative to an absolute number, as an indicator. The value of the degree is found by dividing one by the resulting number.
Rice. 1
Consider the power of a number with a negative fractional exponent. Let's imagine that the number a is any positive number, numbers n And m - integers. According to definition a , which is raised to the power - equals one divided by the same number with positive degree(Figure 1). When the power of a number is a fraction, then in such cases only numbers with positive exponents are used.
Worth remembering that zero can never be an exponent of a number (the rule of division by zero).
The spread of such a concept as a number became such manipulations as measurement calculations, as well as the development of mathematics as a science. The introduction of negative values was due to the development of algebra, which gave general solutions arithmetic problems, regardless of their specific meaning and initial numerical data. In India, back in the 6th-11th centuries, negative numbers were systematically used when solving problems and were interpreted in the same way as today. In European science, negative numbers began to be widely used thanks to R. Descartes, who gave a geometric interpretation of negative numbers as the directions of segments. It was Descartes who proposed the designation of a number raised to a power to be displayed as a two-story formula a n .
In this material we will look at what a power of a number is. In addition to the basic definitions, we will formulate what powers with natural, integer, rational and irrational exponents are. As always, all concepts will be illustrated with example problems.
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First, let's formulate the basic definition of a degree with a natural exponent. To do this, we need to remember the basic rules of multiplication. Let us clarify in advance that for now we will take a real number as a base (denoted by the letter a), and a natural number as an indicator (denoted by the letter n).
Definition 1
The power of a number a with natural exponent n is the product of the nth number of factors, each of which is equal to the number a. The degree is written like this: a n, and in the form of a formula its composition can be represented as follows:
For example, if the exponent is 1 and the base is a, then the first power of a is written as a 1. Given that a is the value of the factor and 1 is the number of factors, we can conclude that a 1 = a.
In general, we can say that a degree is a convenient form of recording large quantity equal factors. So, a record of the form 8 8 8 8 can be shortened to 8 4 . In much the same way, a work helps us avoid recording large number terms (8 + 8 + 8 + 8 = 8 4) ; We have already discussed this in the article devoted to the multiplication of natural numbers.
How to correctly read the degree entry? The generally accepted option is “a to the power of n”. Or you can say “nth power of a” or “anth power”. If, say, in the example we encountered the entry 8 12 , we can read "8 to the 12th power", "8 to the power of 12" or "12th power of 8".
The second and third powers of numbers have their own established names: square and cube. If we see the second power, for example, the number 7 (7 2), then we can say “7 squared” or “square of the number 7”. Similarly, the third degree is read like this: 5 3 - this is the “cube of the number 5” or “5 cubed.” However, you can also use the standard formulation “to the second/third power”; this will not be a mistake.
Example 1
Let's look at an example of a degree with a natural exponent: for 5 7 five will be the base, and seven will be the exponent.
The base does not have to be an integer: for the degree (4 , 32) 9 the base will be the fraction 4, 32, and the exponent will be nine. Pay attention to the parentheses: this notation is made for all powers whose bases differ from natural numbers.
For example: 1 2 3, (- 3) 12, - 2 3 5 2, 2, 4 35 5, 7 3.
What are parentheses for? They help avoid errors in calculations. Let's say we have two entries: (− 2) 3 And − 2 3 . The first of these means a negative number minus two raised to a power with a natural exponent of three; the second is the number corresponding to the opposite value of the degree 2 3 .
Sometimes in books you can find a slightly different spelling of the power of a number - a^n(where a is the base and n is the exponent). That is, 4^9 is the same as 4 9 . If n is a multi-digit number, it is placed in parentheses. For example, 15 ^ (21) , (− 3 , 1) ^ (156) . But we will use the notation a n as more common.
It’s easy to guess how to calculate the value of an exponent with a natural exponent from its definition: you just need to multiply a nth number of times. We wrote more about this in another article.
The concept of degree is the inverse of another mathematical concept - the root of a number. If we know the value of the power and the exponent, we can calculate its base. The degree has some specific properties, useful for solving problems that we discussed in a separate material.
Exponents can include not only natural numbers, but also any integer values in general, including negative ones and zeros, because they also belong to the set of integers.
Definition 2
The power of a number with a positive integer exponent can be represented as a formula: 
.
In this case, n is any positive integer.
Let's understand the concept of zero degree. To do this, we use an approach that takes into account the quotient property for powers with equal bases. It is formulated like this:
Definition 3
Equality a m: a n = a m − n will be true under the following conditions: m and n are natural numbers, m< n , a ≠ 0 .
The last condition is important because it avoids division by zero. If the values of m and n are equal, then we get the following result: a n: a n = a n − n = a 0
But at the same time a n: a n = 1 is the quotient of equal numbers a n and a. It turns out that the zero power of any non-zero number is equal to one.
However, such a proof does not apply to zero to the zeroth power. To do this, we need another property of powers - the property of products of powers with equal bases. It looks like this: a m · a n = a m + n .
If n is equal to 0, then a m · a 0 = a m(this equality also proves to us that a 0 = 1). But if and is also equal to zero, our equality takes the form 0 m · 0 0 = 0 m, It will be true for any natural value of n, and it does not matter what exactly the value of the degree is equal to 0 0 , that is, it can be equal to any number, and this will not affect the accuracy of the equality. Therefore, a notation of the form 0 0 does not have its own special meaning, and we will not attribute it to it.
If desired, it is easy to check that a 0 = 1 converges with the degree property (a m) n = a m n provided that the base of the degree is not zero. Thus, the power of any non-zero number with exponent zero is one.
Example 2
Let's look at an example with specific numbers: So, 5 0 - unit, (33 , 3) 0 = 1 , - 4 5 9 0 = 1 , and the value 0 0 undefined.
After the zero degree, we just have to figure out what a negative degree is. To do this, we need the same property of the product of powers with equal bases that we already used above: a m · a n = a m + n.
Let us introduce the condition: m = − n, then a should not be equal to zero. It follows that a − n · a n = a − n + n = a 0 = 1. It turns out that a n and a−n we have mutually reciprocal numbers.
As a result, a to the negative whole power is nothing more than the fraction 1 a n.
This formulation confirms that for a degree with an integer negative exponent, all the same properties are valid that a degree with a natural exponent has (provided that the base is not equal to zero).
Example 3
A power a with a negative integer exponent n can be represented as a fraction 1 a n . Thus, a - n = 1 a n subject to a ≠ 0 and n – any natural number.
Let us illustrate our idea with specific examples:
Example 4
3 - 2 = 1 3 2 , (- 4 . 2) - 5 = 1 (- 4 . 2) 5 , 11 37 - 1 = 1 11 37 1
In the last part of the paragraph, we will try to depict everything that has been said clearly in one formula:
Definition 4
The power of a number with a natural exponent z is: a z = a z, e with l and z - positive integer 1, z = 0 and a ≠ 0, (for z = 0 and a = 0 the result is 0 0, the values of the expression 0 0 are not is defined) 1 a z, if and z is a negative integer and a ≠ 0 ( if z is a negative integer and a = 0 you get 0 z, egoz the value is undetermined)
What are powers with a rational exponent?
We examined cases when the exponent contains an integer. However, you can raise a number to a power even when its exponent contains a fractional number. This is called degree c rational indicator. In this section we will prove that it has the same properties as other powers.
What's happened rational numbers? Their variety includes both whole and fractional numbers, while fractional numbers can be represented as ordinary fractions (both positive and negative). Let us formulate the definition of the power of a number a with a fractional exponent m / n, where n is a natural number and m is an integer.
We have some degree with a fractional exponent a m n . In order for the power to power property to hold, the equality a m n n = a m n · n = a m must be true.
Given the definition of the nth root and that a m n n = a m, we can accept the condition a m n = a m n if a m n makes sense for the given values of m, n and a.
The above properties of a degree with an integer exponent will be true under the condition a m n = a m n .
The main conclusion from our reasoning is this: the power of a certain number a with a fractional exponent m / n is the nth root of the number a to the power m. This is true if, for given values of m, n and a, the expression a m n remains meaningful.
1. We can limit the value of the base of the degree: let's take a, which for positive values of m will be greater than or equal to 0, and for negative values - strictly less (since for m ≤ 0 we get 0 m, but such a degree is not defined). In this case, the definition of a degree with a fractional exponent will look like this:
A power with a fractional exponent m/n for some positive number a is the nth root of a raised to the power m. This can be expressed as a formula:
For a power with a zero base, this provision is also suitable, but only if its exponent is a positive number.
A power with a base zero and a fractional positive exponent m/n can be expressed as
0 m n = 0 m n = 0 provided m is a positive integer and n is a natural number.
For a negative ratio m n< 0 степень не определяется, т.е. такая запись смысла не имеет.
Let's note one point. Since we introduced the condition that a is greater than or equal to zero, we ended up discarding some cases.
The expression a m n sometimes still makes sense for some negative values of a and some m. Thus, the correct entries are (- 5) 2 3, (- 1, 2) 5 7, - 1 2 - 8 4, in which the base is negative.
2. The second approach is to consider separately the root a m n with even and odd exponents. Then we will need to introduce one more condition: the degree a, in the exponent of which there is a reducible ordinary fraction, is considered to be the degree a, in the exponent of which there is the corresponding irreducible fraction. Later we will explain why we need this condition and why it is so important. Thus, if we have the notation a m · k n · k , then we can reduce it to a m n and simplify the calculations.
If n is an odd number and the value of m is positive and a is any non-negative number, then a m n makes sense. The condition for a to be non-negative is necessary because a root of an even degree cannot be extracted from a negative number. If the value of m is positive, then a can be both negative and zero, because an odd root can be taken from any real number.
Let's combine all the above definitions in one entry:
Here m/n means an irreducible fraction, m is any integer, and n is any natural number.
Definition 5
For any ordinary reducible fraction m · k n · k the degree can be replaced by a m n .
The power of a number a with an irreducible fractional exponent m / n – can be expressed as a m n in the following cases: - for any real a, positive integer values m and odd natural values n. Example: 2 5 3 = 2 5 3, (- 5, 1) 2 7 = (- 5, 1) - 2 7, 0 5 19 = 0 5 19.
For any non-zero real a, negative integer values of m and odd values of n, for example, 2 - 5 3 = 2 - 5 3, (- 5, 1) - 2 7 = (- 5, 1) - 2 7
For any non-negative a, positive integer m and even n, for example, 2 1 4 = 2 1 4, (5, 1) 3 2 = (5, 1) 3, 0 7 18 = 0 7 18.
For any positive a, negative integer m and even n, for example, 2 - 1 4 = 2 - 1 4, (5, 1) - 3 2 = (5, 1) - 3, .
In the case of other values, the degree with a fractional exponent is not determined. Examples of such degrees: - 2 11 6, - 2 1 2 3 2, 0 - 2 5.
Now let’s explain the importance of the condition discussed above: why replace a fraction with a reducible exponent with a fraction with an irreducible exponent. If we had not done this, we would have had the following situations, say, 6/10 = 3/5. Then it should be true (- 1) 6 10 = - 1 3 5 , but - 1 6 10 = (- 1) 6 10 = 1 10 = 1 10 10 = 1 , and (- 1) 3 5 = (- 1) 3 5 = - 1 5 = - 1 5 5 = - 1 .
The definition of a degree with a fractional exponent, which we presented first, is more convenient to use in practice than the second, so we will continue to use it.
Definition 6
Thus, the power of a positive number a with a fractional exponent m/n is defined as 0 m n = 0 m n = 0. In case of negative a the notation a m n does not make sense. Power of zero for positive fractional exponents m/n is defined as 0 m n = 0 m n = 0 , for negative fractional exponents we do not define the degree of zero.
In conclusions, we note that any fractional indicator can be written both in the form of a mixed number and in the form decimal: 5 1 , 7 , 3 2 5 - 2 3 7 .
When calculating, it is better to replace the exponent with an ordinary fraction and then use the definition of exponent with a fractional exponent. For the examples above we get:
5 1 , 7 = 5 17 10 = 5 7 10 3 2 5 - 2 3 7 = 3 2 5 - 17 7 = 3 2 5 - 17 7
What are powers with irrational and real exponents?
What are real numbers? Their set includes both rational and irrational numbers. Therefore, in order to understand what a degree with a real exponent is, we need to define degrees with rational and irrational exponents. We have already mentioned rational ones above. Let's deal with irrational indicators step by step.
Example 5
Let's assume that we have an irrational number a and a sequence of its decimal approximations a 0 , a 1 , a 2 , . . . . For example, let's take the value a = 1.67175331. . . , Then
a 0 = 1, 6, a 1 = 1, 67, a 2 = 1, 671, . . . , a 0 = 1.67, a 1 = 1.6717, a 2 = 1.671753, . . .
We can associate sequences of approximations with a sequence of degrees a a 0 , a a 1 , a a 2 , . . . . If we remember what we said earlier about raising numbers to rational powers, then we can calculate the values of these powers ourselves.
Let's take for example a = 3, then a a 0 = 3 1, 67, a a 1 = 3 1, 6717, a a 2 = 3 1, 671753, . . . etc.
The sequence of powers can be reduced to a number, which will be the value of the power with base a and irrational exponent a. As a result: a degree with an irrational exponent of the form 3 1, 67175331. . can be reduced to the number 6, 27.
Definition 7
The power of a positive number a with an irrational exponent a is written as a a . Its value is the limit of the sequence a a 0 , a a 1 , a a 2 , . . . , where a 0 , a 1 , a 2 , . . . are successive decimal approximations of the irrational number a. A degree with a zero base can also be defined for positive irrational exponents, with 0 a = 0 So, 0 6 = 0, 0 21 3 3 = 0. But this cannot be done for negative ones, since, for example, the value 0 - 5, 0 - 2 π is not defined. A unit raised to any irrational power remains a unit, for example, and 1 2, 1 5 in 2 and 1 - 5 will be equal to 1.
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In one of the previous articles we already mentioned the power of a number. Today we will try to navigate the process of finding its meaning. Scientifically speaking, we will figure out how to raise to a power correctly. We will figure out how this process is carried out, and at the same time we will touch on all possible exponents: natural, irrational, rational, integer.
So, let's take a closer look at the solutions to the examples and find out what it means:
- Definition of the concept.
- Raising to negative art.
- A whole indicator.
- Raising a number to an irrational power.
Here is a definition that accurately reflects the meaning: “Exponentiation is the definition of the value of a power of a number.”
Accordingly, raising the number a in Art. r and the process of finding the value of the degree a with the exponent r are identical concepts. For example, if the task is to calculate the value of the power (0.6)6″, then it can be simplified to the expression “Raise the number 0.6 to the power of 6.”
After this, you can proceed directly to the construction rules.
Raising to a negative power
For clarity, you should pay attention to the following chain of expressions:
110=0.1=1* 10 minus 1 tbsp.,
1100=0.01=1*10 in minus 2 degrees,
11000=0.0001=1*10 in minus 3 st.,
110000=0.00001=1*10 to minus 4 degrees.
Thanks to these examples, you can clearly see the ability to instantly calculate 10 to any minus power. For this purpose, it is enough to simply shift the decimal component:
- 10 to the -1 degree - before one there is 1 zero;
- in -3 - three zeros before one;
- in -9 there are 9 zeros and so on.
It is also easy to understand from this diagram how much 10 minus 5 tbsp will be. -
1100000=0,000001=(1*10)-5.
How to raise a number to a natural power
Remembering the definition, we take into account that the natural number a in Art. n equals the product of n factors, each of which equals a. Let's illustrate: (a*a*…a)n, where n is the number of numbers that are multiplied. Accordingly, in order to raise a to n, it is necessary to calculate the product the following type: a*a*…a divided by n times.
From this it becomes obvious that raising to natural st. relies on the ability to perform multiplication(this material is covered in the section on multiplying real numbers). Let's look at the problem:
Raise -2 to the 4th st.
We are dealing with a natural indicator. Accordingly, the course of the decision will be as follows: (-2) in Art. 4 = (-2)*(-2)*(-2)*(-2). Now all that remains is to multiply the integers: (-2)*(-2)*(-2)*(-2). We get 16.
Answer to the problem:
(-2) in Art. 4=16.
Example:
Calculate the value: three point two sevenths squared.
This example equals the following product: three point two sevenths multiplied by three point two sevenths. Recalling how mixed numbers are multiplied, we complete the construction:
- 3 point 2 sevenths multiplied by themselves;
- equals 23 sevenths multiplied by 23 sevenths;
- equals 529 forty-ninths;
- we reduce and we get 10 thirty-nine forty-ninths.
Answer: 10 39/49
Regarding the issue of raising to an irrational exponent, it should be noted that calculations begin to be carried out after the completion of preliminary rounding of the basis of the degree to any digit that would allow obtaining the value with a given accuracy. For example, we need to square the number P (pi).
We start by rounding P to hundredths and get:
P squared = (3.14)2=9.8596. However, if we reduce P to ten thousandths, we get P = 3.14159. Then squaring gives a completely different number: 9.8695877281.
It should be noted here that in many problems there is no need to raise irrational numbers to powers. As a rule, the answer is entered either in the form of the actual degree, for example, the root of 6 to the power of 3, or, if the expression allows, its transformation is carried out: root of 5 to 7 degrees = 125 root of 5.
How to raise a number to an integer power
This algebraic manipulation is appropriate take into account for the following cases:
- for integers;
- for a zero indicator;
- for a positive integer exponent.
Since almost all are intact positive numbers coincide with the mass of natural numbers, then setting it to a positive integer power is the same process as setting it in Art. natural. We described this process in the previous paragraph.
Now let's talk about calculating st. null. We have already found out above that zero degree numbers a can be determined for any non-zero a (real), while a in Art. 0 will equal 1.
Accordingly, raising any real number to the zero st. will give one.
For example, 10 in st. 0=1, (-3.65)0=1, and 0 in st. 0 cannot be determined.
In order to complete raising to an integer power, it remains to decide on the options for negative integer values. We remember that Art. from a with an integer exponent -z will be defined as a fraction. The denominator of the fraction is st. with the whole positive value, the meaning of which we have already learned to find. Now all that remains is to consider an example of construction.
Example:
Calculate the value of the number 2 cubed with a negative integer exponent.
Solution process:
According to the definition of a degree with a negative exponent, we denote: two minus 3 degrees. equals one to two to the third power.
The denominator is calculated simply: two cubed;
3 = 2*2*2=8.
Answer: two to the minus 3rd art. = one eighth.
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 identical 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.
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