DEGREE WITH A RATIONAL INDICATOR,

POWER FUNCTION IV

§ 79. Extracting roots from a work and a quotient

Theorem 1. Root P th power of the product of positive numbers is equal to the product of the roots P -th degree of the factors, that is, when A > 0, b > 0 and natural P

n √ab = n √a n √b . (1)

Proof. Recall that the root P th power of a positive number ab there is a positive number P -th degree of which is equal to ab . Therefore, proving equality (1) is the same as proving the equality

(n √a n √b ) n = ab .

By the property of the degree of the product

(n √a n √b ) n = (n √a ) n (n √b ) n =.

But by definition of the root P th degree ( n √a ) n = A , (n √b ) n = b .

That's why ( n √a n √b ) n = ab . The theorem has been proven.

Requirement A > 0, b > 0 is essential only for even P , because for negative A And b and even P roots n √a And n √b not defined. If P odd, then formula (1) is valid for any A And b (both positive and negative).

Examples: √16 121 = √16 √121 = 4 11 = 44.

3 √-125 27 = 3 √-125 3 √27 = -5 3 = - 15

Formula (1) is useful when calculating the roots, when the root expression is represented as a product of exact squares. For example,

√153 2 -72 2 = √ (153+ 72) (153-72) = √225 81 = 15 9 = 135.

We proved Theorem 1 for the case when the radical sign on the left side of formula (1) is the product of two positive numbers. In fact, this theorem is true for any number of positive factors, that is, for any natural k > 2:

Consequence. Reading this identity from right to left, we get the following rule for multiplying roots with the same exponents;

To multiply roots with the same exponents, it is enough to multiply the root expressions, leaving the exponent of the root the same.

For example, √3 √8 √6 = √3 8 6 = √144 = 12.

Theorem 2. Root P th power of a fraction whose numerator and denominator are positive numbers is equal to the quotient of dividing the root of the same degree from the numerator by the root of the same degree from the denominator, that is, when A > 0 and b > 0

(2)

To prove equality (2) means to show that

According to the rule of raising a fraction to a power and determining the root n th degree we have:

Thus the theorem is proved.

Requirement A > 0 and b > 0 is essential only for even P . If P odd, then formula (2) is also true for negative values A And b .

Consequence. Reading identity from right to left, we get the following rule for dividing roots with the same exponents:

To divide roots with the same exponents, it is enough to divide the root expressions, leaving the exponent of the root the same.

For example,

Exercises

554. Where in the proof of Theorem 1 did we use the fact that A And b positive?

Why with an odd P formula (1) is also true for negative numbers A And b ?

At what values X the equality data is correct (No. 555-560):

555. √x 2 - 9 = √x -3 √x + 3 .

556. 4 √ (x - 2) (8 - x ) = 4 √x - 2 4 √ 8 - x

557. 3 √ (X + 1) (X - 5) = 3 √x +1 3 √x - 5 .

558. √ X (X + 1) (X + 2) = √ X √ (X + 1) √ (X + 2)

559. √ (x - a ) 3 = (√ x - a ) 3 .

560. 3 √ (X - 5) 2 = (3 √ X - 5 ) 2 .

561. Calculate:

a) √ 173 2 - 52 2 ; V) √ 200 2 - 56 2 ;

b) √ 3732 - 2522; G) √ 242,5 2 - 46,5 2 .

562. In right triangle the hypotenuse is 205 cm, and one of the legs is 84 cm. Find the other leg.

563. How many times:

555. X > 3. 556. 2 < X < 8. 557. X - any number. 558. X > 0. 559. X > A . 560. X - any number. 563. a) Three times.

Subject information: Introduce the square root theorem for fractions. Consolidation of the knowledge gained by students on the topics: “Arithmetic square root”, “Square root of a degree”, “Square root of a product”. Strengthening the skills of quick counting.

Activity-communication: development and formation of students' skills of logical thinking, correct and competent speech, quick reaction.

Value-oriented: arouse students' interest in the study of this topic and this subject. Ability to apply acquired knowledge in practical activities and on other subjects.

1. Repeat the definition of arithmetic square root.

2. Repeat the square root theorem from the degree.

3. Repeat the square root theorem from the product.

4. Develop oral counting skills.

5. Prepare students to study the topic “square root of a fraction” and to master the material of geometry.

6. Tell about the history of the origin of the arithmetic root.

Didactic materials and equipment: didactic lesson map (Appendix 1), blackboard, chalk, cards for individual tasks (taking into account the individual abilities of students), cards for oral counting, cards for independent work.

During the classes:

1. Organizing time: write down the topic of the lesson, setting the goal and objectives of the lesson (for students).

Topic lesson: The square root of a fraction.

The purpose of the lesson: today in the lesson we will repeat the definition of the arithmetic square root, the theorem on the square root of the degree and the square root of the product. And let's get acquainted with the theorem on the square root of a fraction.

Lesson objectives:

1) repeat with the help of mental counting the definitions of the square root and theorems on the square root of the degree and product;

2) during the oral count, some guys will complete tasks on cards;

3) explanation of new material;

4) historical reference;

5) completing tasks independent work(as a test).

2. Frontal survey:

1) verbal counting: take the square root of the following expressions:

a) using the definition of the square root, calculate:;;; ;

b) tabular values: ; ;;;;; ;

c) the square root of the product ;;;;

d) the square root of the degree;;;;; ;

e) take the common factor out of brackets:;; ;.

2) individual work by cards: Annex 2.

3. Check D/Z:

4. Explanation of the new material:

Write a task for students on the board according to the options “calculate the square root of a fraction”:

Option 1: =

Option 2: =

If the guys completed the first task: ask how they did it?

Option 1: presented in the form of a square and received. Make a conclusion.

Option 2: presented the numerator and denominator using the definition of the degree in the form and received.

Give more examples, for example, calculate the square root of a fraction; ; .

Draw an analogy in literal form:

Enter the theorem.

Theorem. If a is greater than or equal to 0, c is greater than 0, then the root of the fraction a / b is equal to the fraction in the numerator of which is the root of a and the denominator is the root of b, i.e. The root of a fraction is equal to the root of the numerator divided by the root of the denominator.

Let us prove that 1) the root of a divided by the root of c is greater than or equal to 0

Proof. 1) Because the root of a is greater than or equal to 0 and the root of c is greater than 0 then the root of a divided by the root of c is greater than or equal to 0.

2)

5. Consolidation of new material: from the textbook of Sh. A. Alimov: No. 362 (1.3); No. 363 (2.3); No. 364 (2.4); №365 (2.3)

6. Historical reference.

The arithmetic root comes from the Latin word radix - root, radicalis - root

Beginning in the 13th century, Italian and other European mathematicians denoted the root with the Latin word radix (abbreviated as r). In 1525, in the book of H. Rudolph "Fast and beautiful counting with the help of skillful rules of algebra, usually called Koss", the designation V for the square root appeared; the cube root was denoted VVV. In 1626, the Dutch mathematician A. Girard introduced the designations V, VV, VVV, etc., which were soon supplanted by the sign r, while a horizontal line was placed above the radical expression. The modern designation of the root first appeared in the book Geometry by René Descartes, published in 1637.

8. Homework: № 362 (2,4); № 363 (1,4); № 364 (1,3); №365 (1,4)

In this article, we will analyze the main root properties. Let's start with the properties of the arithmetic square root, give their formulations and give proofs. After that, we will deal with the properties of the arithmetic root of the nth degree.

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Square root properties

In this section, we will deal with the following main properties of the arithmetic square root:

In each of the written equalities, the left and right parts can be interchanged, for example, equality can be rewritten as . In this "reverse" form, the properties of the arithmetic square root are applied when simplification of expressions just as often as in the "direct" form.

The proof of the first two properties is based on the definition of the arithmetic square root and on . And to justify the last property of the arithmetic square root, you have to remember.

So let's start with proof of the property of the arithmetic square root of the product of two non-negative numbers: . To do this, according to the definition of the arithmetic square root, it suffices to show that is a non-negative number whose square is equal to a b . Let's do it. The value of the expression is non-negative as the product of non-negative numbers. The property of the degree of the product of two numbers allows us to write the equality , and since by the definition of the arithmetic square root and , then .

Similarly, it is proved that the arithmetic square root of the product of k non-negative factors a 1 , a 2 , …, a k is equal to the product of arithmetic square roots from these multipliers. Really, . It follows from this equality that .

Here are some examples: and .

Now let's prove property of the arithmetic square root of a quotient: . The property of the natural power quotient allows us to write the equality , A , while there is a non-negative number. This is the proof.

For example, and .

It's time to disassemble property of the arithmetic square root of the square of a number, in the form of equality it is written as . To prove it, consider two cases: for a≥0 and for a<0 .

It is obvious that for a≥0 the equality is true. It is also easy to see that for a<0 будет верно равенство . Действительно, в этом случае −a>0 and (−a) 2 =a 2 . Thus, , which was to be proved.

Here are some examples: And .

The property of the square root just proved allows us to justify the following result, where a is any real number, and m is any. Indeed, the exponentiation property allows us to replace the degree a 2 m by the expression (a m) 2 , then .

Eg, And .

Properties of the nth root

Let's first list the main properties of nth roots:

All written equalities remain valid if the left and right sides are interchanged in them. In this form, they are also often used, mainly when simplifying and transforming expressions.

The proof of all voiced properties of the root is based on the definition of the arithmetic root of the nth degree, on the properties of the degree and on the definition of the module of the number. Let's prove them in order of priority.

Let's start with the proof properties of the nth root of a product . For non-negative a and b, the value of the expression is also non-negative, as is the product of non-negative numbers. The product property of natural powers allows us to write the equality . By definition of the arithmetic root of the nth degree and, therefore, . This proves the considered property of the root.

This property is proved similarly for the product of k factors: for non-negative numbers a 1 , a 2 , …, a n And .

Here are examples of using the property of the root of the nth degree of the product: And .

Let's prove root property of quotient. For a≥0 and b>0, the condition is satisfied, and .

Let's show examples: And .

We move on. Let's prove property of the nth root of a number to the power of n. That is, we will prove that for any real a and natural m . For a≥0 we have and , which proves the equality , and the equality obviously. For a<0 имеем и (the last transition is valid due to the power property with an even exponent), which proves the equality , and is true due to the fact that when talking about the root of an odd degree, we took for any non-negative number c .

Here are examples of using the parsed root property: and .

We proceed to the proof of the property of the root from the root. Let's swap the right and left parts, that is, we will prove the validity of the equality , which will mean the validity of the original equality. For a non-negative number a, the square root of the form is a non-negative number. Remembering the property of raising a power to a power, and using the definition of the root, we can write a chain of equalities of the form . This proves the considered property of a root from a root.

The property of a root from a root from a root is proved similarly, and so on. Really, .

For example, And .

Let us prove the following root exponent reduction property. To do this, by virtue of the definition of the root, it suffices to show that there is a non-negative number that, when raised to the power of n m, is equal to a m . Let's do it. It is clear that if the number a is non-negative, then the n-th root of the number a is a non-negative number. Wherein , which completes the proof.

Here is an example of using the parsed root property: .

Let us prove the following property, the property of the root of the degree of the form . It is obvious that for a≥0 the degree is a non-negative number. Moreover, its nth power is equal to a m , indeed, . This proves the considered property of the degree.

For example, .

Let's move on. Let us prove that for any positive numbers a and b for which the condition a , that is, a≥b . And this contradicts the condition a

For example, we give the correct inequality .

Finally, it remains to prove the last property of the nth root. Let us first prove the first part of this property, that is, we will prove that for m>n and 0 . Then, due to the properties of a degree with a natural exponent, the inequality , that is, a n ≤ a m . And the resulting inequality for m>n and 0

Similarly, by contradiction, it is proved that for m>n and a>1 the condition is satisfied.

Let us give examples of the application of the proved property of the root in concrete numbers. For example, the inequalities and are true.

Bibliography.

• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8 cells. educational institutions.
• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the Beginnings of Analysis: A Textbook for Grades 10-11 of General Educational Institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for applicants to technical schools).

The square root of a is a number whose square is a. For example, the numbers -5 and 5 are the square roots of the number 25. That is, the roots of the equation x^2=25 are the square roots of the number 25. Now you need to learn how to work with the square root operation: study its basic properties.

The square root of the product

√(a*b)=√a*√b

The square root of the product of two non-negative numbers is equal to the product of the square roots of these numbers. For example, √(9*25) = √9*√25 =3*5 =15;

It is important to understand that this property also applies to the case when the radical expression is the product of three, four, etc. non-negative multipliers.

Sometimes there is another formulation of this property. If a and b are non-negative numbers, then the following equality holds: √(a*b) =√a*√b. There is absolutely no difference between them, you can use either one or the other wording (which one is more convenient to remember).

The square root of a fraction

If a>=0 and b>0, then the following equality is true:

√(a/b)=√a/√b.

For example, √(9/25) = √9/√25 =3/5;

This property also has a different formulation, in my opinion, more convenient to remember.
The square root of the quotient is equal to the quotient of the roots.

It is worth noting that these formulas work both from left to right and from right to left. That is, if necessary, we can represent the product of the roots as the root of the product. The same goes for the second property.

As you can see, these properties are very convenient, and I would like to have the same properties for addition and subtraction:

√(a+b)=√a+√b;

√(a-b)=√a-√b;

But unfortunately such properties are square have no roots, and so cannot be done in calculations..