Addition of numbers with different exponents. Actions with monomials
How to multiply powers? Which powers can be multiplied and which cannot? How do you multiply a number by a power?
In algebra, you can find the product of powers in two cases:
1) if the degrees have the same basis;
2) if the degrees have the same indicators.
When multiplying powers with the same base, the base must remain the same, and the exponents must be added:
When multiplying degrees with the same indicators, the total indicator can be taken out of brackets:
Consider how to multiply powers, with specific examples.
The unit in the exponent is not written, but when multiplying the degrees, they take into account:
When multiplying, the number of degrees can be any. It should be remembered that you can not write the multiplication sign before the letter:
In expressions, exponentiation is performed first.
If you need to multiply a number by a power, you must first perform exponentiation, and only then - multiplication:
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Addition, subtraction, multiplication, and division of powers
Addition and subtraction of powers
Obviously, numbers with powers can be added like other quantities , by adding them one by one 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 the same powers of the same variables can be added or subtracted.
So, the sum of 2a 2 and 3a 2 is 5a 2 .
It is also obvious that if we take two squares a, or three squares a, or five squares a.
But degrees various variables and various degrees identical variables, must be added by adding them to 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 neither twice the square of a, but 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 subtrahend must be changed accordingly.
Or:
2a 4 - (-6a 4) = 8a 4
3h 2 b 6 - 4h 2 b 6 \u003d -h 2 b 6
5(a - h) 6 - 2(a - h) 6 = 3(a - h) 6
Power multiplication
Numbers with powers can be multiplied like other quantities by writing them one after the other, with or without the multiplication sign between them.
So, 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 the same 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 sum 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 is;
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.
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 is equal to the sum or difference of their squares.
If 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 degree.
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 divisor, or by placing them in the form of a fraction.
So a 3 b 2 divided by b 2 is a 3 .
Writing a 5 divided by a 3 looks like $\frac $. 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 powers with the same base, their exponents are subtracted..
So, y 3:y 2 = y 3-2 = y 1 . That is, $\frac = y$.
And a n+1:a = a n+1-1 = a n . That is, $\frac = a^n$.
Or:
y2m: ym = ym
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b + y) n-3
The rule is also valid for numbers with negative degree values.
The result of dividing a -5 by a -3 is a -2 .
Also, $\frac: \frac = \frac .\frac = \frac = \frac $.
h 2:h -1 = h 2+1 = h 3 or $h^2:\frac = h^2.\frac = h^3$
It is necessary to master the 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 exponents in $\frac $ Answer: $\frac $.
2. Reduce the exponents in $\frac$. Answer: $\frac $ or 2x.
3. Reduce the exponents a 2 / a 3 and a -3 / a -4 and bring to common denominator.
a 2 .a -4 is a -2 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.
degree properties
We remind you that in this lesson understand degree properties with natural indicators and zero. Degrees with rational indicators and their properties will be discussed in lessons for grade 8.
Degree c natural indicator has several important properties, which allow you to simplify calculations in examples with powers.
Property #1
Product of powers
When multiplying powers with the same base, the base remains unchanged, and the exponents are added.
a m a n \u003d a m + n, where "a" is any number, and "m", "n" are any natural numbers.
This property of powers also affects the product of three or more powers.
b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15
6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17
(0.8) 3 (0.8) 12 = (0.8) 3 + 12 = (0.8) 15
Please note that in the indicated property it was only about multiplying 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
calculate (3 3 + 3 2) = (27 + 9) = 36 and 3 5 = 243
Property #2
Private 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.
(2b) 5: (2b) 3 = (2b) 5 − 3 = (2b) 2
11 3 - 2 4 2 - 1 = 11 4 = 44
Example. Solve the equation. We use the property of partial degrees.
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 degree properties.
2 11 − 5 = 2 6 = 64
Please note that property 2 dealt only with the division of 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 #3
Exponentiation
When raising a power to a power, the base of the power remains unchanged, and the exponents are multiplied.
(a n) m \u003d a n m, where "a" is any number, and "m", "n" are any natural numbers.
Please note that property No. 4, like other properties of degrees, is also used in reverse order.
(a n b n)= (a b) n
That is, to multiply degrees with the same exponents, you can multiply the bases, and leave the exponent unchanged.
2 4 5 4 = (2 5) 4 = 10 4 = 10,000
0.5 16 2 16 = (0.5 2) 16 = 1
In more difficult examples there may be cases when multiplication and division must be performed on powers with different bases and different exponents. 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
Example of exponentiation of a decimal fraction.
4 21 (−0.25) 20 = 4 4 20 (−0.25) 20 = 4 (4 (−0.25)) 20 = 4 (−1) 20 = 4 1 = four
Properties 5
Power of the quotient (fractions)
To raise a quotient to a power, you can raise the dividend and divisor separately to this power, and divide the first result by the second.
(a: b) n \u003d a n: b n, where "a", "b" are any rational numbers, b ≠ 0, n is any natural number.
(5: 3) 12 = 5 12: 3 12
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.
Degrees and Roots
Operations with powers and roots. Degree with negative ,
zero and fractional indicator. About expressions that don't make sense.
Operations with degrees.
1. When multiplying powers with the same base, their indicators are added up:
a m · a n = a m + n .
2. When dividing degrees with the same base, their indicators subtracted .
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 the 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 degree to a power, their indicators are multiplied:
All of the above formulas are read and executed in both directions from left to right and vice versa.
EXAMPLE (2 3 5 / 15)² = 2 ² 3 ² 5 ² / 15 ² = 900 / 225 = 4 .
Operations with roots. In all the formulas below, the symbol means arithmetic root(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 the ratio is equal to the ratio of the roots of the dividend and divisor:
3. When raising a root to a power, it is enough to raise to this power root number:
4. If you increase the degree of the root by m times and simultaneously raise the root number to the m -th degree, then the value of the root will not change:
5. If you reduce the degree of the root by m times and at the same time extract the root of the m-th degree from the radical number, then the value of the root will not change:
Extension of the concept of degree. So far, we have considered degrees only with a natural indicator; but operations with powers and roots can also lead to negative, zero and fractional indicators. All these exponents require an additional definition.
Degree with a negative exponent. The degree of a certain number with a negative (integer) exponent is defined as one divided by the degree 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 at 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 at m = n, we need a definition of the zero degree.
Degree with zero exponent. The degree of any non-zero number with zero exponent is 1.
EXAMPLES. 2 0 = 1, ( – 5) 0 = 1, (– 3 / 5) 0 = 1.
A degree with a fractional exponent. In order to raise real number and to the power m / n, you need to extract the root of the nth degree from the mth power of this number a:
About expressions that don't make sense. There are several such expressions.
where a ≠ 0 , does not exist.
Indeed, 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.
Indeed, 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 holds for any number x, which was to be proved.
0 0 — any number.
Solution. 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, whence follows,
what x- any number; but taking into account that
our case x> 0 , the answer is x > 0 ;
Rules for multiplying powers with different bases
DEGREE WITH A RATIONAL INDICATOR,
POWER FUNCTION IV
§ 69. Multiplication and division of powers with the same bases
Theorem 1. To multiply powers with the same bases, it is enough to add the exponents, and leave the base the same, that is
Proof. By definition of degree
2 2 2 3 = 2 5 = 32; (-3) (-3) 3 = (-3) 4 = 81.
We have considered the product of two powers. In fact, the proved property is true for any number of powers with the same bases.
Theorem 2. To divide powers with the same bases, when the indicator of the dividend is greater than the indicator of the divisor, it is enough to subtract the indicator of the divisor from the indicator of the dividend, and leave the base the same, that is at t > n
(a =/= 0)
Proof. Recall that the quotient of dividing one number by another is the number that, when multiplied by a divisor, gives the dividend. Therefore, prove the formula , where a =/= 0, it's like proving the formula
If a t > n , then the number t - p will be natural; therefore, by Theorem 1
Theorem 2 is proved.
Note that the formula
proved by us only under the assumption that t > n . Therefore, from what has been proved, it is not yet possible to draw, for example, the following conclusions:
In addition, we have not yet considered degrees with negative exponents, and we do not yet know what meaning can be given to the expression 3 - 2 .
Theorem 3. To raise a power to a power, it is enough to multiply the exponents, leaving the base of the exponent the same, that is
Proof. Using the definition of degree and Theorem 1 of this section, we get:
Q.E.D.
For example, (2 3) 2 = 2 6 = 64;
518 (Oral.) Determine X from the equations:
1) 2 2 2 2 3 2 4 2 5 2 6 = 2 x ; 3) 4 2 4 4 4 6 4 8 4 10 = 2 x ;
2) 3 3 3 3 5 3 7 3 9 = 3 x ; 4) 1 / 5 1 / 25 1 / 125 1 / 625 = 1 / 5 x .
519. (Adjusted) Simplify:
520. (Adjusted) Simplify:
521. Present these expressions as degrees with the same bases:
1) 32 and 64; 3) 85 and 163; 5) 4 100 and 32 50;
2) -1000 and 100; 4) -27 and -243; 6) 81 75 8 200 and 3 600 4 150.
The concept of a degree in mathematics is introduced as early as the 7th grade in an algebra lesson. And in the future, throughout the course of studying mathematics, this concept is actively used in its various forms. Degrees are a rather difficult topic, requiring memorization of values and the ability to correctly and quickly count. For faster and better work with mathematics degrees, they came up with the properties of a degree. They help to cut down on big calculations, to convert a huge example into a single number to some extent. There are not so many properties, and all of them are easy to remember and apply in practice. Therefore, the article discusses the main properties of the degree, as well as where they are applied.
degree properties
We will consider 12 properties of a degree, including properties of powers with the same base, and give an example for each property. Each of these properties will help you solve problems with degrees faster, as well as save you from numerous computational errors.
1st property.
Many people very often forget about this property, make mistakes, representing a number to the zero degree as zero.
2nd property.
3rd property.
It must be remembered that this property can only be used when multiplying numbers, it does not work with the sum! And we must not forget that this and the following properties apply only to powers with the same base.
4th property.
If the number in the denominator is raised to a negative power, then when subtracting, the degree of the denominator is taken in brackets to correctly replace the sign in further calculations.
The property only works when dividing, not when subtracting!
5th property.
6th property.
This property can also be applied to reverse side. A unit divided by a number to some degree is that number to a negative power.
7th property.
This property cannot be applied to sum and difference! When raising a sum or difference to a power, abbreviated multiplication formulas are used, not the properties of the power.
8th property.
9th property.
This property works for any fractional degree with a numerator equal to one, the formula will be the same, only the degree of the root will change depending on the denominator of the degree.
Also, this property is often used in reverse order. The root of any power of a number can be represented as that number to the power of one divided by the power of the root. This property is very useful in cases where the root of the number is not extracted.
10th property.
This property works not only with square root and second degree. If the degree of the root and the degree to which this root is raised are the same, then the answer will be a radical expression.
11th property.
You need to be able to see this property in time when solving it in order to save yourself from huge calculations.
12th property.
Each of these properties will meet you more than once in tasks, it can be given in its pure form, or it may require some transformations and the use of other formulas. Therefore, for the correct solution, it is not enough to know only the properties, you need to practice and connect the rest of mathematical knowledge.
Application of degrees and their properties
They are actively used in algebra and geometry. Degrees in mathematics have a separate, important place. With their help, exponential equations and inequalities are solved, as well as powers often complicate equations and examples related to other sections of mathematics. Exponents help to avoid large and long calculations, it is easier to reduce and calculate the exponents. But to work with large powers, or with powers of large numbers, you need to know not only the properties of the degree, but also competently work with the bases, be able to decompose them in order to make your task easier. For convenience, you should also know the meaning of numbers raised to a power. This will reduce your time in solving by eliminating the need for long calculations.
The concept of degree plays a special role in logarithms. Since the logarithm, in essence, is the power of a number.
Abbreviated multiplication formulas are another example of the use of powers. They cannot use the properties of degrees, they are decomposed according to special rules, but in each abbreviated multiplication formula there are invariably degrees.
Degrees are also actively used in physics and computer science. All translations into the SI system are made using degrees, and in the future, when solving problems, the properties of the degree are applied. In computer science, powers of two are actively used, for the convenience of counting and simplifying the perception of numbers. Further calculations for conversions of units of measurement or calculations of problems, just as in physics, occur using the properties of the degree.
Degrees are also very useful in astronomy, where you can rarely find the use of the properties of a degree, but the degrees themselves are actively used to shorten the recording of various quantities and distances.
Degrees are also used in ordinary life, when calculating areas, volumes, distances.
With the help of degrees, very large and very small values \u200b\u200bare written in any field of science.
exponential equations and inequalities
Special place degree properties occupy precisely in exponential equations and inequalities. These tasks are very common, as in school course as well as in exams. All of them are solved by applying the properties of the degree. The unknown is always in the degree itself, therefore, knowing all the properties, it will not be difficult to solve such an equation or inequality.
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 look at the most important questions concerning 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.
We will understand how to solve various problems using this value. 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 the construction 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 computing in more difficult 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 building a work 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 a negative number raised to an odd power, 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 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.
Obviously, numbers with powers can be added like other quantities , by adding them one by one 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 the same powers of the same variables can be added or subtracted.
So, the sum of 2a 2 and 3a 2 is 5a 2 .
It is also obvious that if we take two squares a, or three squares a, or five squares a.
But degrees various variables and various degrees identical variables, must be added by adding them to 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 neither twice the square of a, but 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 subtrahend 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
Power multiplication
Numbers with powers can be multiplied like other quantities by writing them one after the other, with or without the multiplication sign between them.
So, 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 the same 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 sum 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 is;
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.
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 is equal to the sum or difference of their squares.
If 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 degree.
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 divisor, or by placing them in the form of a fraction.
So a 3 b 2 divided by b 2 is 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 powers 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:
y2m: ym = ym
8a n+m: 4a m = 2a n
12(b + y) n: 3(b + y) 3 = 4(b + y) n-3
The rule is also valid for numbers with negative degree values.
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 the 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 in $\frac(5a^4)(3a^2)$ Answer: $\frac(5a^2)(3)$.
2. Reduce the exponents in $\frac(6x^6)(3x^5)$. 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 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.
If you need to raise a specific number to a power, you can use . We will now take a closer look at properties of degrees.
Exponential numbers open up great possibilities, they allow us to convert multiplication into addition, and addition is much easier than multiplication.
For example, we need to multiply 16 by 64. The product of multiplying these two numbers is 1024. But 16 is 4x4, and 64 is 4x4x4. So 16 times 64=4x4x4x4x4 which is also 1024.
The number 16 can also be represented as 2x2x2x2, and 64 as 2x2x2x2x2x2, and if we multiply, we again get 1024.
Now let's use the rule. 16=4 2 , or 2 4 , 64=4 3 , or 2 6 , while 1024=6 4 =4 5 , or 2 10 .
Therefore, our problem can be written in another way: 4 2 x4 3 =4 5 or 2 4 x2 6 =2 10, and each time we get 1024.
We can solve a number of similar examples and see that the multiplication of numbers with powers reduces to addition of exponents, or an exponent, of course, provided that the bases of the factors are equal.
Thus, we can, without multiplying, immediately say that 2 4 x2 2 x2 14 \u003d 2 20.
This rule is also true when dividing numbers with powers, but in this case, e the exponent of the divisor is subtracted from the exponent of the dividend. Thus, 2 5:2 3 =2 2 , which in ordinary numbers is 32:8=4, that is, 2 2 . Let's summarize:
a m x a n \u003d a m + n, a m: a n \u003d a m-n, where m and n are integers.
At first glance, it might seem that multiplication and division of numbers with powers not very convenient, because first you need to represent the number in exponential form. It is not difficult to represent the numbers 8 and 16 in this form, that is, 2 3 and 2 4, but how to do this with the numbers 7 and 17? Or what to do in those cases when the number can be represented in exponential form, but the bases of exponential expressions of numbers are very different. For example, 8×9 is 2 3 x 3 2 , in which case we cannot sum the exponents. Neither 2 5 nor 3 5 is the answer, nor is the answer between the two.
Then is it worth bothering with this method at all? Definitely worth it. It provides huge advantages, especially for complex and time-consuming calculations.