\ (\ sqrt (a) = b \) if \ (b ^ 2 = a \), where \ (a≥0, b≥0 \)

Examples:

\ (\ sqrt (49) = 7 \) since \ (7 ^ 2 = 49 \)
\ (\ sqrt (0.04) = 0.2 \) since \ (0.2 ^ 2 = 0.04 \)

How do you extract the square root of a number?

To extract the square root of a number, you need to ask yourself the question: what number in the square will the expression under the root give?

For example... 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 number squared will give \ (\ sqrt (1 \ frac (13) (36)) \)? To answer 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 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 the opposite. Addition has subtraction, and multiplication has division. The inverse of squaring is to 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:

Of course, when working with a square root, you need to use others as well.

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 retrieved

Example: \ (\ sqrt (2) \), \ (\ sqrt (53) \), \ (\ sqrt (200) \), \ (\ sqrt (0,1) \) etc. - it is not always possible to extract the root from a number and this is normal!

Root of a number, also a number

It is not necessary to refer to \ (\ sqrt (2) \), \ (\ sqrt (53) \), somehow especially. These are numbers, but not whole numbers, yes, but not everything in our world is measured in whole numbers.

The root is extracted only from non-negative numbers

Therefore, in the textbooks you will not see such entries \ (\ sqrt (-23) \), \ (\ sqrt (-1) \), etc.

Title: Independent and test work in algebra and geometry for grade 8.

The manual contains independent and control works on all the most important topics of the course of algebra and geometry in grade 8.

The works consist of 6 variants of three levels of difficulty. Didactic materials are intended for the organization of differentiated independent work of students.

CONTENT
ALGEBRA 4
P-1 Rational Expression. Reducing fractions 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 Rational Expression Transformation 12
С-5 Inverse proportionality and its graph 14
К-2 Rational fractions 16
C-6 Arithmetic square root of 18
C-7 Equation x2 = a. Function y = y [x 20
С-8 Square root of a product, fraction, power 22
K-3 Arithmetic square root and its properties 24
C-9 Introduction and removal of a multiplier in square roots 27
C-10 Converting expressions containing square roots 28
K-4 Applying the properties of the arithmetic square root 30
P-11 Incomplete Quadratic Equations 32
С-12 The formula for the roots of a quadratic equation 33
С-13 Problem solving using quadratic equations. Vieta's theorem 34
K-5 Quadratic Equations 36
P-14 Fractional Rational Equations 38
С-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
С-19 Degree with negative indicator 52
K-9 Degree with integer 54
К-10 Annual test 56
GEOMETRY (According to Pogorelov) 58
С-1 Properties and signs of a parallelogram. "58
C-2 Rectangle. Rhombus. Square 60
K-1 Parallelogram 62
С-3 Thales' theorem. Midline of triangle 63
C-4 Trapezium. Middle line of trapezoid 66
K-2 Trapezium. Midlines of a triangle and a trapezoid ... 68
C-5 Pythagorean Theorem 70
С-6 The opposite theorem to the Pythagorean theorem. Perpendicular and Oblique 71
C-7 Triangle inequality 73
K-3 Pythagorean Theorem 74
C-8 Right Triangle Solution 76
C-9 Properties of trigonometric functions 78
К-4 Right-angled triangle (generalizing test) 80
С-10 Coordinates of the segment midpoint. Distance between points. Equation of the 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
S-13. Parallel transfer 90
С-14 Vector concept. Equality of vectors 92
С-15 Actions with vectors in coordinate form. Collinear vectors 94
С-16 Actions with vectors in geometric form 95
C-17 Dot product 98
K-6 Vectors 99
К-7 Annual examination 102
GEOMETRY (According to Atanasyan) 104
С-1 Properties and signs of a parallelogram 104
C-2 Rectangle. Rhombus. Square 106
К-1 Quadrangles 108
С-3 Area of ​​a rectangle, square 109
С-4 Area of ​​a parallelogram, rhombus, triangle 111
С-5 Trapezium 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
С-9 Applying similarity to problem solving 124
C-10 Relationship between the sides and corners of a right triangle 126
К-4 Application of similarity to problem solving. Ratios between the sides and angles of a right triangle 128
С-11 Tangent to circle 130
С-12 Center and inscribed corners 132
С-13 Theorem on the product of segments of intersecting chords. Wonderful Points of Triangle 134
С-14 Inscribed and circumscribed circles 136
K-5 Circumference 137
C-15 Vector addition and subtraction 139
С-16 Multiplication of a vector by the number 141
С-17 Middle line of trapezoid 142
K-6 Vectors. Applying Vectors to Problem Solving 144
К-7 Annual examination 146
ANSWERS 148
REFERENCES 157

FOREWORD .
1. One relatively small book contains a full set of tests (including final tests) for the entire course of 8th grade algebra and geometry, so it is enough to purchase one set of books per class.
Test papers 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 test work reflects its subject matter.

2. The collection allows for a differentiated control of knowledge, since the tasks are distributed over 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 with an increased interest in mathematics, and also for use in classrooms, schools, grammar schools and high schools with advanced study of mathematics. For each level, there are 2 adjacent equivalent options (as they are usually written on the board), so one book on the desk is enough for the lesson.

Free download an e-book in a convenient format, watch and read:
Download the book Independent work and tests in algebra and geometry for grade 8. Ershova A.P., Goloborodko V.V., 2004 - fileskachat.com, fast and free download.

• Independent and control work in geometry for grade 11. Goloborodko V.V., Ershova A.P., 2004
• Independent and test work in algebra and geometry for the 9th grade. Ershova A.P., Goloborodko V.V., 2004
• Independent and control works on algebra and geometry, grade 8, Ershova A.P., Goloborodko V.V., Ershova A.S., 2013

I looked again at the sign ... And let's go!

Let's start with a simple one:

Just a minute. this, which means that we can write like this:

Got it? Here's the next one for you:

The roots of the resulting numbers are not exactly extracted? It doesn't matter - here are some examples:

But what if the factors are not two, but more? Same! The root multiplication formula works with any number of factors:

Now completely on its own:

Answers: Well done! Agree, everything is very easy, the main thing is to know the multiplication table!

Division of roots

We figured out the multiplication of roots, now we will proceed to the property of division.

Let me remind you that the general formula looks like this:

This means that the root of the quotient is equal to the quotient of the roots.

Well, let's figure it out with examples:

That's all science. Here's an example:

Everything is not as smooth as in the first example, but, as you can see, there is nothing complicated.

But what if an expression like this comes across:

You just need to apply the formula in the opposite direction:

And here's an example:

You can also come across this expression:

Everything is the same, only here you need to remember how to translate fractions (if you don't remember, look into the topic and come back!). Remembered? Now we decide!

I am sure that you have coped with everything, everything, now let's try to build roots in power.

Exponentiation

What happens if the square root is squared? It's simple, let's remember the meaning of the square root of a number - this is a number whose square root is equal to.

So, if we raise a number whose square root is equal to the square, then what do we get?

Well, of course, !

Let's look at examples:

It's simple, right? And if the root is in a different degree? It's OK!

Stick to the same logic and remember the properties and possible actions with degrees.

Read the theory on the topic "" and everything will become very 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 of a number to a power? For example, this is:

Pretty simple, right? And if the degree is more than two? We follow the same logic using degree properties:

Well, is everything clear? Then solve the examples yourself:

And here are the answers:

Introduction under the root sign

What have we not learned to do with roots! It remains only to practice entering the number under the root sign!

It's easy!

Let's say we have written down the number

What can we do with it? Well, of course, hide the three under the root, remembering that the three is the square root of!

Why do we need this? Yes, just to expand our capabilities when solving examples:

How do you like this property of roots? Does it make life much easier? For me, that's right! Only we must remember that we can only introduce positive numbers under the square root sign.

Solve this example yourself -
Did you manage? Let's see what you should get:

Well done! You managed to insert the number under the root sign! Let's move on to an equally important one - let's look at how to compare numbers containing the square root!

Comparison of roots

Why should we learn to compare numbers containing the square root?

Very simple. Often, in large and lengthy expressions found on the exam, we get an irrational answer (do you remember what it is? You and I have already talked about this today!)

We need to place the received answers on a coordinate line, for example, to determine which interval is suitable for solving the equation. And here a snag arises: there is no calculator on the exam, and without it how to imagine which number is greater and which is less? That's just it!

For example, define which is greater: or?

You can't tell right off the bat. Well, let's use the analyzed property of entering a number under the root sign?

Then go ahead:

And, obviously, the larger the number under the root sign, the larger the root itself!

Those. if, then,.

From this we firmly conclude that. And no one will convince us otherwise!

Extracting roots from large numbers

Before that, we introduced the factor under the root sign, but how to get it out? You just have to factor it and extract what is extracted!

It was possible to take a different path and decompose into other factors:

Not bad, huh? Any of these approaches is correct, decide what suits you best.

Factoring is very useful when solving non-standard tasks like this:

We are not afraid, but we act! Let us decompose each factor under the root into separate factors:

Now try it yourself (without a calculator! It won't be on the exam):

Is this the end? Don't stop halfway!

That's all, not so scary, right?

Happened? Well done, that's right!

Now try to solve this example:

And an example is a tough nut to crack, so you just can't figure out how to approach it. But we, of course, can tough it.

Well, let's start factoring? Note right away that you can divide a number by (remember the divisibility criteria):

Now, try it yourself (again, without a calculator!):

Well, did it work? Well done, that's right!

Let's summarize

1. The square root (arithmetic square root) of a non-negative number is a non-negative number whose square is equal to.
   .
2. If we just take the square root of something, we always get one non-negative result.
3. Arithmetic root properties:
4. When comparing square roots, it must be remembered that the larger the number under the root sign, 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 on the square root exam.

Now it's your turn. Write to us whether it is a difficult topic for you or not.

Did you learn something new or everything was already clear.

Write in the comments and good luck on your exams!