Division of numbers with large powers. Lesson "multiplication and division of degrees". Multiplication rules for powers with different radix

In the previous article, we described what monomials are. In this article, we will analyze how to solve examples and problems in which they are applied. Here will be considered such actions as subtraction, addition, multiplication, division of monomials and raising them to a power with a natural exponent. We will show how such operations are defined, outline the basic rules for their implementation and what should be the result. All theoretical positions, as usual, will be illustrated with examples of problems with descriptions of solutions.

It is most convenient to work with the standard notation of monomials, therefore, all expressions that will be used in the article are presented in a standard form. If initially they are set differently, it is recommended to first bring them to the generally accepted form.

Addition and Subtraction Rules for Monomials

The simplest operations that can be performed with monomials are subtraction and addition. In the general case, the result of these actions will be a polynomial (a monomial is possible in some special cases).

When we add or subtract monomials, we first write down the corresponding sum and difference in conventional form, and then we simplify the resulting expression. If there are such terms, they need to be given, brackets - to open. Let us explain with an example.

Example 1

Condition: carry out the addition of monomials - 3 x and 2, 72 x 3 y 5 z.

Solution

Let's write down the sum of the original expressions. Add parentheses and put a plus between them. We get the following:

(- 3 x) + (2, 72 x 3 y 5 z)

When we expand the parentheses, we get - 3 · x + 2.72 · x 3 · y 5 · z. This is a polynomial written in standard form, which will be the result of the addition of these monomials.

Answer:(- 3 x) + (2, 72 x 3 y 5 z) = - 3 x + 2, 72 x 3 y 5 z.

If we have given three, four or more terms, we carry out this action in the same way.

Example 2

Condition: carry out the indicated actions with the polynomials in the correct order

3 a 2 - (- 4 a c) + a 2 - 7 a 2 + 4 9 - 2 2 3 a c

Solution

Let's start by expanding the parentheses.

3 a 2 + 4 a c + a 2 - 7 a 2 + 4 9 - 2 2 3 a c

We see that the resulting expression can be simplified by reducing similar terms:

3 a 2 + 4 a c + a 2 - 7 a 2 + 4 9 - 2 2 3 a c = = (3 a 2 + a 2 - 7 a 2) + 4 a c - 2 2 3 a c + 4 9 = = - 3 a 2 + 1 1 3 a c + 4 9

We have got a polynomial, which will be the result of this action.

Answer: 3 a 2 - (- 4 a c) + a 2 - 7 a 2 + 4 9 - 2 2 3 a c = - 3 a 2 + 1 1 3 a c + 4 9

In principle, we can add and subtract two monomials with some restrictions so that we end up with a monomial. To do this, you need to comply with some conditions concerning the summands and subtracted monomials. We will describe how this is done in a separate article.

Monomial multiplication rules

The multiplication action does not impose any restrictions on the multipliers. The monomials to be multiplied do not have to meet any additional conditions for the result to be a monomial.

To perform multiplication of monomials, you need to follow these steps:

1. Record the piece correctly.
2. Expand parentheses in the resulting expression.
3. Group, if possible, factors with the same variables and numerical factors separately.
4. Perform the necessary actions with the numbers and apply the property of multiplying powers with the same bases to the remaining factors.

Let's see how this is done in practice.

Example 3

Condition: multiply the monomials 2 x 4 y z and - 7 16 t 2 x 2 z 11.

Solution

Let's start by compiling a work.

We open the brackets in it and get the following:

2 x 4 y z - 7 16 t 2 x 2 z 11

2 - 7 16 t 2 x 4 x 2 y z 3 z 11

All we have to do is multiply the numbers in the first brackets and apply the power property to the second. As a result, we get the following:

2 - 7 16 t 2 x 4 x 2 y z 3 z 11 = - 7 8 t 2 x 4 + 2 y z 3 + 11 = = - 7 8 t 2 x 6 y z 14

Answer: 2 x 4 y z - 7 16 t 2 x 2 z 11 = - 7 8 t 2 x 6 y z 14.

If we have three or more polynomials in our condition, we multiply them using exactly the same algorithm. We will consider in more detail the question of multiplication of monomials within the framework of a separate material.

Rules for raising a monomial to a power

We know that a degree with a natural exponent is the product of a certain number of identical factors. Their number is indicated by the number in the indicator. According to this definition, raising a monomial to a power is equivalent to multiplying the specified number of identical monomials. Let's see how this is done.

Example 4

Condition: raise the monomial - 2 a b 4 to the power of 3.

Solution

We can replace exponentiation with multiplication of 3 monomials - 2 · a · b 4. Let's write down and get the desired answer:

(- 2 a b 4) 3 = (- 2 a b 4) (- 2 a b 4) (- 2 a b 4) = = ((- 2) (- 2) (- 2)) (a a a) (b 4 b 4 b 4) = - 8 a 3 b 12

Answer:(- 2 a b 4) 3 = - 8 a 3 b 12.

But what if the degree has a large indicator? Writing a large number of factors is inconvenient. Then, to solve such a problem, we need to apply the properties of the degree, namely the property of the degree of the product and the property of the degree in degree.

Let's solve the problem that we gave above in the indicated way.

Example 5

Condition: perform the erection - 2 · a · b 4 to the third power.

Solution

Knowing the property of the degree to the degree, we can proceed to an expression of the following form:

(- 2 a b 4) 3 = (- 2) 3 a 3 (b 4) 3.

After that we raise to the power - 2 and apply the property of the power to the power:

(- 2) 3 (a) 3 (b 4) 3 = - 8 a 3 b 4 3 = - 8 a 3 b 12.

Answer:- 2 a b 4 = - 8 a 3 b 12.

We also devoted a separate article to the raising of a monomial to a power.

Division rules for monomials

The last action with monomials, which we will analyze in this material, is the division of a monomial by a monomial. As a result, we should get a rational (algebraic) fraction (in some cases it is possible to obtain a monomial). Let us clarify right away that division by a zero monomial is not defined, since division by 0 is not defined.

To perform division, we need to write the indicated monomials in the form of a fraction and reduce it, if possible.

Example 6

Condition: divide the monomial - 9 · x 4 · y 3 · z 7 by - 6 · p 3 · t 5 · x 2 · y 2.

Solution

Let's start by writing monomials in fractional form.

9 x 4 y 3 z 7 - 6 p 3 t 5 x 2 y 2

This fraction can be abbreviated. After performing this action, we get:

3 x 2 y z 7 2 p 3 t 5

Answer:- 9 x 4 y 3 z 7 - 6 p 3 t 5 x 2 y 2 = 3 x 2 y z 7 2 p 3 t 5.

The conditions under which, as a result of dividing monomials, we obtain a monomial are given in a separate article.

If you notice an error in the text, please select it and press Ctrl + Enter

If you need to raise a specific number to a power, you can use. And now we will dwell in more detail on properties of degrees.

Exponential numbers open up great possibilities, they allow us to transform multiplication into addition, and adding is much easier than multiplying.

For example, we need to multiply 16 by 64. The product of the multiplication of these two numbers is 1024. But 16 is 4x4, and 64 is 4x4x4. That is, 16 by 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, at the same time 1024 = 6 4 = 4 5, or 2 10.

Therefore, our problem can be written differently: 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 multiplying numbers with powers reduces to addition of exponents, or exponential, of course, provided that the bases of the factors are equal.

Thus, without multiplying, we can immediately say that 2 4 x2 2 x2 14 = 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 х a n = a m + n, a m: a n = a m-n, where m and n are integers.

At first glance, it may seem what is 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 when the number can be represented in exponential form, but the bases of the exponential expressions of numbers are very different. For example, 8 × 9 is 2 3 × 3 2, in which case we cannot sum the exponents. Neither 2 5 nor 3 5 is the answer, nor does the answer lie in the interval between these two numbers.

Then is it worth bothering with this method at all? Definitely worth it. It offers tremendous benefits, especially for complex and time consuming computations.

Apart from the eighth degree, what do we see here? We recall the 7th grade program. So, remember? This is the formula for abbreviated multiplication, namely the difference of squares! We get:

We carefully look at the denominator. It looks a lot like one of the multipliers in the numerator, but what's wrong? Wrong order of terms. If they were to be reversed, the rule could be applied.

But how to do that? It turns out to be very easy: an even degree of the denominator helps us here.

The terms are magically reversed. This "phenomenon" is applicable to any expression to an even degree: we can freely change the signs in brackets.

But it's important to remember: all signs change at the same time!

Let's go back to the example:

And again the formula:

Whole we call the natural numbers opposite to them (that is, taken with the sign "") and the number.

positive integer, but it is no different from natural, then everything looks exactly like in the previous section.

Now let's look at some new cases. Let's start with an indicator equal to.

Any number in the zero degree is equal to one:

As always, let us ask ourselves the question: why is this so?

Consider some degree with a base. Take, for example, and multiply by:

So, we multiplied the number by, and we got the same as it was -. And what number should you multiply so that nothing changes? That's right, on. Means.

We can do the same with an arbitrary number:

Let's repeat the rule:

Any number in the zero degree is equal to one.

But there are exceptions to many rules. And here it is also there - this is a number (as a base).

On the one hand, it should be equal to any degree - no matter how much you multiply by yourself, you will still get zero, this is clear. But on the other hand, like any number in the zero degree, it must be equal. So which of this is true? Mathematicians decided not to get involved and refused to raise zero to zero. That is, now we cannot not only divide by zero, but also raise it to a zero power.

Let's go further. In addition to natural numbers and numbers, negative numbers belong to integers. To understand what a negative power is, let's do the same as last time: multiply some normal number by the same negative power:

From here it is already easy to express what you are looking for:

Now we extend the resulting rule to an arbitrary degree:

So, let's formulate a rule:

A number in the negative power is inverse to the same number in the positive power. But at the same time the base cannot be null:(because you cannot divide by).

Let's summarize:

I. Expression not specified in case. If, then.

II. Any number to the zero degree is equal to one:.

III. A number that is not zero is in negative power inverse to the same number in a positive power:.

Tasks for independent solution:

Well, and, as usual, examples for an independent solution:

Analysis of tasks for independent solution:

I know, I know, the numbers are terrible, but on the exam you have to be ready for anything! Solve these examples or analyze their solution if you could not solve them and you will learn how to easily cope with them on the exam!

Let's continue to expand the circle of numbers "suitable" as an exponent.

Now consider rational numbers. What numbers are called rational?

Answer: all that can be represented as a fraction, where and are integers, moreover.

To understand what is Fractional degree, consider the fraction:

Let's raise both sides of the equation to the power:

Now let's remember the rule about "Degree to degree":

What number must be raised to a power to get?

This formulation is the definition of the th root.

Let me remind you: the root of the th power of a number () is a number that, when raised to a power, is equal to.

That is, the root of the th power is the inverse operation of the exponentiation:.

It turns out that. Obviously, this particular case can be extended:.

Now we add the numerator: what is it? The answer is easily obtained using the degree-to-degree rule:

But can the base be any number? After all, the root can not be extracted from all numbers.

None!

Remember the rule: any number raised to an even power is a positive number. That is, you cannot extract roots of an even degree from negative numbers!

And this means that such numbers cannot be raised to a fractional power with an even denominator, that is, the expression does not make sense.

What about expression?

But this is where the problem arises.

The number can be represented as other, cancellable fractions, for example, or.

And it turns out that it exists, but does not exist, but these are just two different records of the same number.

Or another example: once, then you can write. But if we write down the indicator in a different way, and again we get a nuisance: (that is, we got a completely different result!).

To avoid such paradoxes, we consider only positive radix with fractional exponent.

So if:

• — natural number;
• - an integer;

Examples:

Rational exponents are very useful for converting rooted expressions, for example:

5 examples to train

Analysis of 5 examples for training

1. Do not forget about the usual properties of degrees:

2.. Here we remember that we forgot to learn the table of degrees:

after all - it is or. The solution is found automatically:.

And now the hardest part. Now we will analyze irrational grade.

All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception of

Indeed, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are whole numbers (that is, irrational numbers are all real numbers except rational ones).

When studying degrees with a natural, whole and rational indicator, each time we made up a kind of "image", "analogy", or description in more familiar terms.

For example, a natural exponent is a number multiplied by itself several times;

...zero-degree number- it is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared - therefore, the result is only a kind of "blank number", namely the number;

...integer negative exponent- it is as if a certain " reverse process”, That is, the number was not multiplied by itself, but divided.

By the way, in science, a degree with a complex indicator is often used, that is, the indicator is not even a real number.

But at school we do not think about such difficulties, you will have the opportunity to comprehend these new concepts at the institute.

WHERE WE ARE SURE YOU GO! (if you learn how to solve such examples :))

For example:

Decide for yourself:

Analysis of solutions:

1. Let's start with the already usual rule for raising a power to a power:

Now look at the indicator. Does he remind you of anything? We recall the formula for abbreviated multiplication, the difference of squares:

In this case,

It turns out that:

Answer: .

2. We bring fractions in exponents to the same form: either both decimal, or both ordinary. Let's get, for example:

Answer: 16

3. Nothing special, we apply the usual properties of the degrees:

ADVANCED LEVEL

Determination of the degree

A degree is an expression of the form:, where:

• — base of degree;
• - exponent.

Degree with natural exponent (n = 1, 2, 3, ...)

Raising a number to a natural power n means multiplying the number by itself times:

Integer degree (0, ± 1, ± 2, ...)

If the exponent is whole positive number:

Erection to zero degree:

The expression is indefinite, because, on the one hand, to any degree - this, and on the other - any number to the th degree - this.

If the exponent is whole negative number:

(because you cannot divide by).

Once again about zeros: expression is undefined in case. If, then.

Examples:

Rational grade

• - natural number;
• - an integer;

Examples:

Power properties

To make it easier to solve problems, let's try to understand: where did these properties come from? Let's prove them.

Let's see: what is and?

A-priory:

So, on the right side of this expression, we get the following product:

But by definition, it is the power of a number with an exponent, that is:

Q.E.D.

Example : Simplify the expression.

Solution : .

Example : Simplify the expression.

Solution : It is important to note that in our rule necessarily must have the same bases. Therefore, we combine the degrees with the base, but remains a separate factor:

Another important note: this rule is - only for the product of degrees!

By no means should I write that.

Just as with the previous property, let us turn to the definition of the degree:

Let's rearrange this piece like this:

It turns out that the expression is multiplied by itself once, that is, according to the definition, this is the th power of the number:

In essence, this can be called "bracketing the indicator". But you should never do this in total:!

Let's remember the abbreviated multiplication formulas: how many times did we want to write? But this is not true, after all.

A degree with a negative base.

Up to this point, we have only discussed how it should be index degree. But what should be the foundation? In degrees with natural indicator the basis can be any number .

Indeed, we can multiply any numbers by each other, be they positive, negative, or even. Let's think about which signs ("" or "") will have powers of positive and negative numbers?

For example, will the number be positive or negative? A? ?

With the first, everything is clear: no matter how many positive numbers we multiply by each other, the result will be positive.

But negative is a little more interesting. After all, we remember a simple rule from the 6th grade: "minus by minus gives a plus." That is, or. But if we multiply by (), we get -.

And so on to infinity: with each subsequent multiplication, the sign will change. It is possible to formulate such simple rules:

1. even degree, - number positive.
2. Negative number raised to odd degree, - number negative.
3. A positive number to any degree is a positive number.
4. Zero to any power is equal to zero.

Determine for yourself which sign the following expressions will have:

1.	2.	3.
4.	5.	6.

Did you manage? Here are the answers:

1) ; 2) ; 3) ; 4) ; 5) ; 6) .

In the first four examples, I hope everything is clear? We just look at the base and exponent and apply the appropriate rule.

In example 5), everything is also not as scary as it seems: it does not matter what the base is equal to - the degree is even, which means that the result will always be positive. Well, unless the base is zero. The foundation is not equal, is it? Obviously not, since (because).

Example 6) is no longer so simple. Here you need to find out which is less: or? If you remember that, it becomes clear that, which means that the base is less than zero. That is, we apply rule 2: the result will be negative.

And again we use the definition of degree:

Everything is as usual - we write down the definition of degrees and, divide them into each other, divide them into pairs and get:

Before examining the last rule, let's solve a few examples.

Calculate the values ​​of the expressions:

Solutions :

Apart from the eighth degree, what do we see here? We recall the 7th grade program. So, remember? This is the formula for abbreviated multiplication, namely the difference of squares!

We get:

We carefully look at the denominator. It looks a lot like one of the multipliers in the numerator, but what's wrong? Wrong order of terms. If they were swapped, Rule 3 could be applied. But how can this be done? It turns out to be very easy: an even degree of the denominator helps us here.

If you multiply it by, nothing changes, right? But now it turns out the following:

The terms are magically reversed. This "phenomenon" is applicable to any expression to an even degree: we can freely change the signs in brackets. But it's important to remember: all signs change at the same time! It cannot be replaced with by changing only one disadvantage that we do not like!

Let's go back to the example:

And again the formula:

So now the last rule:

How are we going to prove it? Of course, as usual: let's expand the concept of degree and simplify:

Now let's open the brackets. How many letters will there be? times by multipliers - what does it look like? This is nothing more than a definition of an operation multiplication: there were only multipliers. That is, it is, by definition, the degree of a number with an exponent:

Example:

Irrational grade

In addition to the information about the degrees for the intermediate level, here is the degree with an irrational exponent. All the rules and properties of degrees here are exactly the same as for a degree with a rational exponent, with the exception - after all, by definition, irrational numbers are numbers that cannot be represented as a fraction, where and are whole numbers (that is, irrational numbers are all real numbers except rational).

When studying degrees with a natural, whole and rational indicator, each time we made up a kind of "image", "analogy", or description in more familiar terms. For example, a natural exponent is a number multiplied by itself several times; a number to the zero degree is, as it were, a number multiplied by itself once, that is, it has not yet begun to be multiplied, which means that the number itself has not even appeared yet - therefore, the result is only a kind of "blank number", namely the number; a degree with an integer negative exponent is as if some kind of "reverse process" took place, that is, the number was not multiplied by itself, but divided.

It is extremely difficult to imagine a degree with an irrational exponent (just as it is difficult to imagine a 4-dimensional space). Rather, it is a purely mathematical object that mathematicians created to extend the concept of a degree to the entire space of numbers.

By the way, in science, a degree with a complex indicator is often used, that is, the indicator is not even a real number. But at school we do not think about such difficulties, you will have the opportunity to comprehend these new concepts at the institute.

So what do we do when we see an irrational exponent? We are trying with all our might to get rid of it! :)

For example:

Decide for yourself:

1)

2)

3)

Answers:

1. We recall the formula for the difference of squares. Answer: .
2. We bring fractions to the same form: either both decimal places, or both ordinary ones. We get, for example:.
3. Nothing special, we apply the usual degree properties:

SUMMARY OF THE SECTION AND BASIC FORMULAS

Degree is called an expression of the form:, where:

Integer degree

degree, the exponent of which is a natural number (i.e. whole and positive).

Rational grade

degree, the exponent of which is negative and fractional numbers.

Irrational grade

degree whose exponent is infinite decimal or root.

Power properties

Features of degrees.

• Negative number raised to even degree, - number positive.
• Negative number raised to odd degree, - number negative.
• A positive number to any degree is a positive number.
• Zero is equal to any degree.
• Any number to the zero degree is equal to.

NOW YOUR WORD ...

How do you like the article? Write down in the comments like whether you like it or not.

Tell us about your experience with degree properties.

Perhaps you have questions. Or suggestions.

Write in the comments.

And good luck with your exams!

We remind you that this lesson understands power properties with natural indicators and zero. Rational degrees and their properties will be covered in the 8th grade lessons.

A natural exponent has several important properties that make it easier to calculate in exponent examples.

Property number 1
Product of degrees

Remember!

When multiplying degrees with the same bases, the base remains unchanged, and the exponents are added.

a m · a n = a m + n, where "a" is any number, and "m", "n" are any natural numbers.

This property of degrees also affects the product of three or more degrees.

• Simplify the expression.
  b b 2 b 3 b 4 b 5 = b 1 + 2 + 3 + 4 + 5 = b 15
• Present as a degree.
  6 15 36 = 6 15 6 2 = 6 15 6 2 = 6 17
• Present as a degree.
  (0.8) 3 (0.8) 12 = (0.8) 3 + 12 = (0.8) 15

Important!

Please note that in the specified property it was only about the multiplication of powers with on the same grounds ... It does not apply to their addition.

You cannot replace the amount (3 3 + 3 2) with 3 5. This is understandable if
count (3 3 + 3 2) = (27 + 9) = 36, and 3 5 = 243

Property number 2
Private degrees

Remember!

When dividing degrees with the same bases, 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 private degrees.
3 8: t = 3 4

T = 3 8 - 4

Answer: t = 3 4 = 81

Using properties # 1 and # 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 the degree.
  = = = 2 9 + 2

  2 5

  = 2 11

  2 5

  = 2 11 − 5 = 2 6 = 64

  Important!

  Note that property 2 was only about dividing degrees with the same bases.

  You cannot replace the difference (4 3 −4 2) with 4 1. This is understandable if you count (4 3 −4 2) = (64 − 16) = 48 , and 4 1 = 4

  Be careful!

  Property number 3
  Exponentiation

  Remember!

  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.

  
  

  Properties 4
  Degree of work

  Remember!

  When raising to the power of a product, each of the factors is raised to a power. The results are then 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 s) 2 = 6 2 a 2 2 b 3 2 s 1 2 = 36 a 4 b 6 s 2
    
  • Example 2.
    (−x 2 y) 6 = ((−1) 6 x 2 6 y 1 6) = x 12 y 6

  Important!

  Note that property # 4, like other degree properties, is applied in reverse order.

  (a n b n) = (a b) n

  That is, in order to multiply degrees with the same indicators, you can multiply the bases, and the exponent can be left unchanged.

  • 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

  In more complex examples, there may be cases when multiplication and division must be performed over degrees with different bases and different exponents. In this case, we advise you to proceed as follows.

  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 to a decimal 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
  Degree of quotient (fraction)

  Remember!

  To raise a quotient to a power, you can raise a separate dividend and a divisor 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 is any natural number.

  • Example. Present the expression in the form of private degrees.
    (5: 3) 12 = 5 12: 3 12

  We remind you that the quotient can be represented as a fraction. Therefore, on the topic raising a fraction to a power we will go into more detail on the next page.