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 the arithmetic square roots of these factors. 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 . In this way, 
, 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 
.
For example, 
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 instance, 
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 instance, 
.
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 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.
, that is, a n ≤ a m . And the resulting inequality for m>n and 0
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).
I looked again at the plate ... And, let's go!
Let's start with a simple one:
Wait a minute. this, which means we can write it like this:
Got it? Here's the next one for you:
The roots of the resulting numbers are not exactly extracted? Don't worry, here are some examples:
But what if there are not two multipliers, but more? The same! The root multiplication formula works with any number of factors:
Now completely independent:
Answers: Well done! Agree, everything is very easy, the main thing is to know the multiplication table!
Root division
We figured out the multiplication of the roots, now let's proceed to the property of division.
Let me remind you that the formula in general looks like this:
And that means that the root of the quotient is equal to the quotient of the roots.
Well, let's look at examples:
That's all science. And here's an example:
Everything is not as smooth as in the first example, but as you can see, there is nothing complicated.
What if the expression looks like this:
You just need to apply the formula in reverse:
And here's an example:
You can also see this expression:
Everything is the same, only here you need to remember how to translate fractions (if you don’t remember, look at the topic and come back!). Remembered? Now we decide!
I am sure that you coped with everything, everything, now let's try to build roots in a degree.
Exponentiation
What happens if the square root is squared? It's simple, remember the meaning of the square root of a number - this is a number whose square root is equal to.
So, if we square a number whose square root is equal, then what do we get?
Well, of course, !
Let's look at examples:
Everything is simple, right? And if the root is in a different degree? Nothing wrong!
Stick to the same logic and remember the properties and possible actions with degrees.
Read the theory on the topic "" and everything will become extremely clear to you.
For example, here's an expression:
In this example, the degree is even, but what if it is odd? Again, apply the power properties and factor everything:
With this, everything seems to be clear, but how to extract the root from a number in a degree? Here, for example, is this:
Pretty simple, right? What if the degree is greater than two? We follow the same logic using the properties of degrees:
Well, is everything clear? Then solve your own examples:
And here are the answers:
Introduction under the sign of the root
What we just have not learned to do with the roots! It remains only to practice entering the number under the root sign!
It's quite easy!
Let's say we have a number
What can we do with it? Well, of course, hide the triple under the root, while remembering that the triple is the square root of!
Why do we need it? Yes, just to expand our capabilities when solving examples:
How do you like this property of roots? Makes life much easier? For me, that's right! Only we must remember that we can only enter positive numbers under the square root sign.
Try this example for yourself:
Did you manage? Let's see what you should get:
Well done! You managed to enter a number under the root sign! Let's move on to something equally important - consider how to compare numbers containing a square root!
Root Comparison
Why should we learn to compare numbers containing a square root?
Very simple. Often, in large and long expressions encountered in the exam, we get an irrational answer (do you remember what it is? We already talked about this today!)
We need to place the received answers on the coordinate line, for example, to determine which interval is suitable for solving the equation. And this is where the snag arises: there is no calculator on the exam, and without it, how to imagine which number is larger and which is smaller? That's it!
For example, determine which is greater: or?
You won't say right off the bat. Well, let's use the parsed property of adding a number under the root sign?
Then forward:
Well, obviously, the larger the number under the sign of the root, the larger the root itself!
Those. if means .
From this we firmly conclude that And no one will convince us otherwise!
Extracting roots from large numbers
Before that, we introduced a factor under the sign of the root, but how to take it out? You just need to factor it out and extract what is extracted!
It was possible to go the other way and decompose into other factors:
Not bad, right? Any of these approaches is correct, decide how you feel comfortable.
Factoring is very useful when solving such non-standard tasks as this one:
We don't get scared, we act! We decompose each factor under the root into separate factors:
And now try it yourself (without a calculator! It will not be on the exam):
Is this the end? We don't stop halfway!
That's all, it's not all that scary, right?
Happened? Well done, you're right!
Now try this example:
And an example is a tough nut to crack, so you can’t immediately figure out how to approach it. But we, of course, are in the teeth.
Well, let's start factoring, shall we? Immediately, we note that you can divide a number by (recall the signs of divisibility):
And now, try it yourself (again, without a calculator!):
Well, did it work? Well done, you're right!
Summing up
- The square root (arithmetic square root) of a non-negative number is a non-negative number whose square is equal.
. - If we just take the square root of something, we always get one non-negative result.
- Arithmetic root properties:
- When comparing square roots, it must be remembered that the larger the number under the sign of the root, the larger the root itself.
How do you like the square root? All clear?
We tried to explain to you without water everything you need to know in the exam about the square root.
Now your turn. Write to us whether this topic is difficult for you or not.
Did you learn something new or everything was already so clear.
Write in the comments and good luck on the exams!
\(\sqrt(a)=b\) if \(b^2=a\), where \(a≥0,b≥0\)
Examples:
\(\sqrt(49)=7\) because \(7^2=49\)
\(\sqrt(0.04)=0.2\),because \(0.2^2=0.04\)
How to extract the square root of a number?
To extract the square root of a number, you need to ask yourself the question: what number squared will give the expression under the root?
for instance. Extract the root: a)\(\sqrt(2500)\); b) \(\sqrt(\frac(4)(9))\); c) \(\sqrt(0.001)\); d) \(\sqrt(1\frac(13)(36))\)
a) What number squared will give \(2500\)?
\(\sqrt(2500)=50\)
b) What number squared will give \(\frac(4)(9)\) ?
\(\sqrt(\frac(4)(9))\) \(=\)\(\frac(2)(3)\)
c) What number squared will give \(0.0001\)?
\(\sqrt(0.0001)=0.01\)
d) What squared number will \(\sqrt(1\frac(13)(36))\) give? To give an answer to the question, you need to translate into the wrong one.
\(\sqrt(1\frac(13)(36))=\sqrt(\frac(49)(16))=\frac(7)(6)\)
Comment: Although \(-50\), \(-\frac(2)(3)\) , \(-0,01\),\(- \frac(7)(6)\) also answer the given questions, but they are not taken into account, since the square root is always positive.
The main property of the root
As you know, in mathematics, any action has an inverse. Addition has subtraction, multiplication has division. The opposite of squaring is taking the square root. Therefore, these actions cancel each other out:
\((\sqrt(a))^2=a\)
This is the main property of the root, which is most often used (including in the OGE)
Example . (task from the OGE). Find the value of the expression \(\frac((2\sqrt(6))^2)(36)\)
Solution :\(\frac((2\sqrt(6))^2)(36)=\frac(4 \cdot (\sqrt(6))^2)(36)=\frac(4 \cdot 6)(36 )=\frac(4)(6)=\frac(2)(3)\)
Example . (task from the OGE). Find the value of the expression \((\sqrt(85)-1)^2\)
Solution:
Answer: \(86-2\sqrt(85)\)Of course, when working with a square root, you need to use others.
Example
. (task from the OGE). Find the value of the expression \(5\sqrt(11) \cdot 2\sqrt(2)\cdot \sqrt(22)\)
Solution:
Answer: \(220\)
4 rules that are always forgotten
The root is not always extracted
Example: \(\sqrt(2)\),\(\sqrt(53)\),\(\sqrt(200)\),\(\sqrt(0,1)\) etc. - extracting a root from a number is not always possible and this is normal!
Root of a number, also a number
No need to treat \(\sqrt(2)\), \(\sqrt(53)\) in any special way. These are numbers, but not integers, yes, but not everything in our world is measured in integers.
The root is taken only from non-negative numbers
Therefore, in textbooks you will not see such entries \(\sqrt(-23)\),\(\sqrt(-1)\), etc.
Title: Independent and control work in algebra and geometry for grade 8.
The manual contains independent and control work on all the most important topics of the 8th grade algebra and geometry course.
The works consist of 6 variants of three levels of difficulty. Didactic materials are designed to organize differentiated independent work of students.
CONTENT
ALGEBRA 4
C-1 Rational expression. Fraction reduction 4
C-2 Adding and subtracting fractions 5
K-1 Rational fractions. Adding and subtracting fractions 7
C-3 Multiplication and division of fractions. Raising a fraction to the power of 10
C-4 Transformation of rational expressions 12
C-5 Inverse proportionality and its plot 14
K-2 Rational fractions 16
C-6 Arithmetic square root 18
C-7 Equation x2 = a. Function y = y[x 20
C-8 Square root of product, fraction, power of 22
K-3 Arithmetic square root and its properties 24
C-9 Insertion and multiplication in square roots 27
C-10 Converting Expressions Containing Square Roots 28
K-4 Application of the properties of the arithmetic square root 30
C-11 Incomplete quadratic equations 32
C-12 Quadratic Root Formula 33
С-13 Problem solving using quadratic equations. Vieta's theorem 34
K-5 Quadratic Equations 36
C-14 Fractional rational equations 38
C-15 Application of fractional rational equations. Problem solving 39
K-6 Fractional Rational Equations 40
C-16 Properties of numerical inequalities 43
K-7 Numerical inequalities and their properties 44
С-17 Linear inequalities with one variable 47
С-18 Systems of linear inequalities 48
K-8 Linear inequalities and systems of inequalities with one variable 50
C-19 Degree with a negative indicator 52
K-9 Degree with integer exponent 54
K-10 Annual test 56
GEOMETRY (According to Pogorelov) 58
C-1 Properties and features of a parallelogram". 58
C-2 Rectangle. Rhombus. Square 60
K-1 Parallelogram 62
C-3 Thales' theorem. Middle line of triangle 63
C-4 Trapeze. Middle line of the trapezoid 66
K-2 Trapeze. Median lines of a triangle and a trapezoid .... 68
C-5 Pythagorean Theorem 70
С-6 Theorem, converse to the Pythagorean theorem. Perpendicular and oblique 71
C-7 Triangle Inequality 73
K-3 Pythagorean theorem 74
C-8 Solving Right Triangles 76
C-9 Properties of trigonometric functions 78
K-4 Right triangle (summary test) 80
С-10 Coordinates of the middle of the segment. Distance between points. Equation of a circle 82
C-11 Equation of a straight line 84
K-5 Cartesian coordinates 86
С-12 Movement and its properties. Central and axial symmetry. turn 88
C-13. Parallel Transfer 90
C-14 The concept of a vector. Vector equality 92
C-15 Operations with vectors in coordinate form. Collinear vectors 94
C-16 Operations with vectors in geometric form 95
C-17 Dot product 98
K-6 Vectors 99
K-7 Annual test 102
GEOMETRY (According to Atanasyan) 104
C-1 Properties and features of a parallelogram 104
C-2 Rectangle. Rhombus. Square 106
K-1 Quadrangles 108
C-3 Area of a rectangle, square 109
C-4 Area of parallelogram, rhombus, triangle 111
C-5 Trapezoid area 113
C-6 Pythagorean theorem 114
K-2 Squares. Pythagorean theorem 116
C-7 Definition of similar triangles. Angle bisector property of a triangle 118
С-8 Signs of similarity of triangles 120
K-3 Similarity of triangles 122
C-9 Applying Similarity to Problem Solving 124
C-10 Relations between sides and angles of a right triangle 126
K-4 Application of similarity to problem solving. Relations between sides and angles of a right triangle 128
C-11 Tangent to circle 130
C-12 Central and inscribed angles 132
C-13 Theorem on the product of segments of intersecting chords. Remarkable Triangle Points 134
C-14 Inscribed and circumscribed circles 136
K-5 Circle 137
C-15 Vector addition and subtraction 139
C-16 Vector multiplication by the number 141
C-17 Middle line of the trapezium 142
K-6 Vectors. Applying vectors to problem solving 144
K-7 Annual test 146
ANSWERS 148
LITERATURE 157
FOREWORD
.
1. One relatively small book contains a complete set of test papers (including final tests) for the entire course of algebra and geometry of the 8th grade, so it is enough to purchase one set of books per class.
Examinations are designed for a lesson, independent work - for 20-35 minutes, depending on the topic. For the convenience of using the book, the title of each independent and control work reflects its subject matter.
2. The collection allows for a differentiated control of knowledge, since the tasks are divided into three levels of complexity A, B and C. Level A corresponds to the mandatory program requirements, B - to the average level of complexity, level C tasks are intended for students who show an increased interest in mathematics, and also for use in classrooms, schools, gymnasiums and lyceums with in-depth study of mathematics. For each level, 2 equivalent options are given next to each other (as they are usually written on the board), so one book per desk is enough for the lesson.
Free download e-book in a convenient format, watch and read:
Download the book Independent and test work in algebra and geometry for grade 8. Ershova A.P., Goloborodko V.V., 2004 - fileskachat.com, fast and free download.
- Independent and control work on geometry for grade 11. Goloborodko V.V., Ershova A.P., 2004
- Independent and control work in algebra and geometry for grade 9. Ershova A.P., Goloborodko V.V., 2004
- Independent and control work in algebra and geometry, grade 8, Ershova A.P., Goloborodko V.V., Ershova A.S., 2013