First level
Root and its properties. Detailed theory with examples (2019)
Let's try to figure out what this concept of “root” is and “what it is eaten with.” To do this, let's look at examples that you have already encountered in class (well, or you are just about to encounter this).
For example, we have an equation. What is the solution to this equation? What numbers can be squared and obtained? Remembering the multiplication table, you can easily give the answer: and (after all, when two negative numbers are multiplied, a positive number is obtained)! To simplify, mathematicians introduced the special concept of the square root and assigned it a special symbol.
Let us define the arithmetic square root.
Why does the number have to be non-negative? For example, what is it equal to? Well, well, let's try to pick one. Maybe three? Let's check: , not. Maybe, ? Again, we check: . Well, it doesn’t fit? This is to be expected - because there are no numbers that, when squared, give a negative number!
This is what you need to remember: the number or expression under the root sign must be non-negative!
However, the most attentive ones have probably already noticed that the definition says that the solution to the square root of “a number is called this non-negative number whose square is equal to ". Some of you will say that at the very beginning we analyzed the example, selected numbers that can be squared and obtained, the answer was and, but here we are talking about some kind of “non-negative number”! This remark is quite appropriate. Here you just need to distinguish between the concepts of quadratic equations and the arithmetic square root of a number. For example, is not equivalent to the expression.
It follows that, that is, or. (Read the topic "")
And it follows that.
Of course, this is very confusing, but it is necessary to remember that the signs are the result of solving the equation, since when solving the equation we must write down all the x's, which, when substituted into the original equation, will give correct result. In our quadratic equation suitable for both.
However, if just take the square root from something, then always we get one non-negative result.
Now try to solve this equation. Everything is not so simple and smooth anymore, is it? Try going through the numbers, maybe something will work out? Let's start from the very beginning - from scratch: - doesn't fit, move on - less than three, also sweep aside, what if. Let's check: - also not suitable, because... that's more than three. It's the same story with negative numbers. So what should we do now? Did the search really give us nothing? Not at all, now we know for sure that the answer will be some number between and, as well as between and. Also, obviously the solutions won't be integers. Moreover, they are not rational. So, what is next? Let's graph the function and mark the solutions on it.
Let's try to cheat the system and get the answer using a calculator! Let's get the root out of it! Oh-oh-oh, it turns out that. This number never ends. How can you remember this, since there won’t be a calculator on the exam!? Everything is very simple, you don’t need to remember it, you just need to remember (or be able to quickly estimate) the approximate value. and the answers themselves. Such numbers are called irrational; it was to simplify the writing of such numbers that the concept of a square root was introduced.
Let's look at another example to reinforce this. Let's look at this problem: you need to cross diagonally square field with a side of km, how many km do you have to walk?
The most obvious thing here is to consider the triangle separately and use the Pythagorean theorem: . Thus, . So what is the required distance here? Obviously, the distance cannot be negative, we get that. The root of two is approximately equal, but, as we noted earlier, - is already a complete answer.
To solve examples with roots without causing problems, you need to see and recognize them. To do this, you need to know at least the squares of numbers from to, and also be able to recognize them. For example, you need to know what is equal to a square, and also, conversely, what is equal to a square.
Did you catch what a square root is? Then solve some examples.
Examples.
Well, how did it work out? Now let's look at these examples:
Answers:
Cube root
Well, we seem to have sorted out the concept of a square root, now let’s try to figure out what a cube root is and what is their difference.
The cube root of a number is the number whose cube is equal to. Have you noticed that everything is much simpler here? There are no restrictions on the possible values of both the value under the cube root sign and the number being extracted. That is, the cube root can be extracted from any number: .
Do you understand what a cube root is and how to extract it? Then go ahead and solve the examples.
Examples.
Answers:
Root - oh degree
Well, we have understood the concepts of square and cube roots. Now let’s summarize the knowledge gained with the concept 1st root.
1st root of a number is a number whose th power is equal, i.e.
equivalent.
If - even, That:
- with negative, the expression does not make sense (even-th roots of negative numbers cannot be removed!);
- for non-negative() expression has one non-negative root.
If - is odd, then the expression has a unique root for any.
Don't be alarmed, the same principles apply here as with square and cube roots. That is, the principles that we applied when considering square roots, extend to all roots of even degree.
And the properties that were used for the cubic root apply to roots of odd degree.
Well, has it become clearer? Let's look at examples:
Here everything is more or less clear: first we look - yeah, the degree is even, the number under the root is positive, which means our task is to find a number whose fourth power will give us. Well, any guesses? Maybe, ? Exactly!
So, the degree is equal - odd, the number under the root is negative. Our task is to find a number that, when raised to a power, produces. It is quite difficult to immediately notice the root. However, you can immediately narrow your search, right? Firstly, the required number is definitely negative, and secondly, one can notice that it is odd, and therefore the desired number is odd. Try to find the root. Of course, you can safely dismiss it. Maybe, ?
Yes, this is what we were looking for! Note that to simplify the calculation we used the properties of degrees: .
Basic properties of roots
It's clear? If not, then after looking at the examples, everything should fall into place.
Multiplying roots
How to multiply roots? The simplest and most basic property helps answer this question:
Let's start with something simple:
Are the roots of the resulting numbers not exactly extracted? No problem - here are some examples:
What if there are not two, but more multipliers? The same! The formula for multiplying roots works with any number of factors:
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 exactly right! You just have to remember that We can only enter positive numbers under the root sign of an even degree.
Let's see where else this can be useful. For example, the problem requires comparing two numbers:
That more:
You can’t tell right away. Well, let's use the disassembled property of entering a number under the root sign? Then go ahead:
Well, knowing what larger number under the sign of the root, the larger the root itself! Those. if, then, . From this we firmly conclude that. And no one will convince us otherwise!
Before this, we entered a multiplier under the sign of the root, but how to remove it? You just need to factor it into factors and extract what you extract!
It was possible to take a different path and expand into other factors:
Not bad, right? Any of these approaches is correct, decide as you wish.
For example, here is an expression:
In this example, the degree is even, but what if it is odd? Again, apply the properties of exponents and factor everything:
Everything seems clear with this, but how to extract the root of a number to a power? 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 here's an example:
These are the pitfalls, about them always worth remembering. This is actually reflected in the property examples:
| for odd: for even and: |
It's clear? Reinforce with examples:
Yeah, we see that the root is to an even power, the negative number under the root is also to an even power. Well, does it work out the same? Here's what:
That's all! Now here are some examples:
Got it? Then go ahead and solve the examples.
Examples.
Answers.
If you have received answers, then you can move on with peace of mind. If not, then let's understand these examples:
Let's look at two other properties of roots:
These properties must be analyzed in examples. Well, let's do this?
Got it? Let's secure it.
Examples.
Answers.
ROOTS AND THEIR PROPERTIES. AVERAGE LEVEL
Arithmetic square root
The equation has two solutions: and. These are numbers whose square is equal to.
Consider the equation. Let's solve it graphically. Let's draw a graph of the function and a line at the level. The intersection points of these lines will be the solutions. We see that this equation also has two solutions - one positive, the other negative:
But in in this case solutions are not integers. Moreover, they are not rational. In order to write down these irrational decisions, we introduce a special square root symbol.
Arithmetic square root is a non-negative number whose square is equal to. When the expression is not defined, because There is no number whose square is equal to a negative number.
Square root: .
For example, . And it follows that or.
Let me draw your attention once again, this is very important: Square root is always a non-negative number: !
Cube root of a number is a number whose cube is equal to. The cube root is defined for everyone. It can be extracted from any number: . As you can see, it can also take negative values.
The th root of a number is a number whose th power is equal, i.e.
If it is even, then:
- if, then the th root of a is not defined.
- if, then the non-negative root of the equation is called the arithmetic root of the th degree of and is denoted.
If - is odd, then the equation has a unique root for any.
Have you noticed that to the left above the sign of the root we write its degree? But not for the square root! If you see a root without a degree, it means it is square (degrees).
Examples.
Basic properties of roots
ROOTS AND THEIR PROPERTIES. BRIEFLY ABOUT THE MAIN THINGS
Square root (arithmetic square root) from a non-negative number is called this non-negative number whose square is
Properties of roots:
Root degree n from a real number a, Where n - natural number, such a real number is called x, n the th degree of which is equal to a.
Root degree n from the number a is indicated by the symbol. According to this definition.
Finding the root n-th degree from among a called root extraction. Number A is called a radical number (expression), n- root indicator. For odd n there is a root n-th power for any real number a. When even n there is a root n-th power only for non-negative numbers a. To disambiguate the root n-th degree from among a, the concept of an arithmetic root is introduced n-th degree from among a.
The concept of an arithmetic root of degree N
If n- natural number, greater 1 , then there is, and only one, non-negative number X, such that the equality is satisfied. This number X called an arithmetic root n th power of a non-negative number A and is designated . Number A is called a radical number, n- root indicator.
So, according to the definition, the notation , where , means, firstly, that and, secondly, that, i.e. .
The concept of a degree with a rational exponent
Degree with natural exponent: let A is a real number, and n- a natural number greater than one, n-th power of the number A call the work n factors, each of which is equal A, i.e. . Number A- the basis of the degree, n- exponent. A power with a zero exponent: by definition, if , then . Zero power of a number 0 doesn't make sense. A degree with a negative integer exponent: assumed by definition if and n is a natural number, then . A degree with a fractional exponent: it is assumed by definition if and n- natural number, m is an integer, then .
Operations with roots.
In all the formulas below, the symbol means an arithmetic root (the 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 a ratio is equal to the ratio of the roots of the dividend and the divisor:
3. When raising a root to a power, it is enough to raise the radical number to this power: 
4. If you increase the degree of the root n times and at the same time raise the radical number to the nth power, then the value of the root will not change: 
5. If you reduce the degree of the root by n times and simultaneously extract the nth root of the radical number, then the value of the root will not change:
Expanding the concept of degree. So far we have considered degrees only with natural exponents; but operations with powers and roots can also lead to negative, zero and fractional exponents. All these exponents require additional definition.
A degree with a negative exponent. The power of a certain number with a negative (integer) exponent is defined as one divided by the power 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 greater than n, but also for 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 to be valid for m = n, we need a definition of degree zero.
A degree with a zero index. The power of any non-zero number with exponent zero is 1.
EXAMPLES. 2 0 = 1, (– 5) 0 = 1, (– 3 / 5) 0 = 1.
Degree with a fractional exponent. In order to raise a real number a to the power m / n, you need to extract the nth root of the mth power of this number a:
About expressions that have no meaning. There are several such expressions.
Case 1.
Where a ≠ 0 does not exist.
In fact, 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
Case 2.
Any number.
In fact, if we assume that this expression is equal to a certain 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 is what needed to be proven.
Really,
Solution. Let's consider three main cases:
1) x = 0 – this value does not satisfy this equation
2) for x > 0 we get: x / x = 1, i.e. 1 = 1, which means that x is any number; but taking into account that in our case x > 0, the answer is x > 0;
3) at x< 0 получаем: – x / x = 1, т.e. –1 = 1, следовательно,
in this case there is no solution. Thus x > 0.
Arithmetic root of the second degree
Definition 1
The second root (or square root) of $a$ call a number that, when squared, becomes equal to $a$.
Example 1
$7^2=7 \cdot 7=49$, which means the number $7$ is the 2nd root of the number $49$;
$0.9^2=0.9 \cdot 0.9=0.81$, which means the number $0.9$ is the 2nd root of the number $0.81$;
$1^2=1 \cdot 1=1$, which means the number $1$ is the 2nd root of the number $1$.
Note 2
Simply put, for any number $a
$a=b^2$ for negative $a$ is incorrect, because $a=b^2$ cannot be negative for any value of $b$.
It can be concluded that For real numbers there cannot be a 2nd root of a negative number.
Note 3
Because $0^2=0 \cdot 0=0$, then from the definition it follows that zero is the 2nd root of zero.
Definition 2
Arithmetic root of the 2nd degree of the number $a$($a \ge 0$) is a non-negative number that, when squared, equals $a$.
Roots of the 2nd degree are also called square roots.
The arithmetic root of the 2nd degree of the number $a$ is denoted as $\sqrt(a)$ or you can see the notation $\sqrt(a)$. But most often for the square root the number $2$ is root exponent– not specified. The sign “$\sqrt( )$” is the sign of the arithmetic root of the 2nd degree, which is also called “ radical sign" The concepts “root” and “radical” are names of the same object.
If there is a number under the arithmetic root sign, then it is called radical number, and if the expression, then – radical expression.
The entry $\sqrt(8)$ is read as “arithmetic root of the 2nd degree of eight,” and the word “arithmetic” is often not used.
Definition 3
According to definition arithmetic root of the 2nd degree can be written:
For any $a \ge 0$:
$(\sqrt(a))^2=a$,
$\sqrt(a)\ge 0$.
We showed the difference between a second root and an arithmetic second root. Further we will consider only roots of non-negative numbers and expressions, i.e. only arithmetic.
Arithmetic root of the third degree
Definition 4
Arithmetic root of the 3rd degree (or cube root) of the number $a$($a \ge 0$) is a non-negative number that, when cubed, becomes equal to $a$.
Often the word arithmetic is omitted and they say “the 3rd root of the number $a$”.
The arithmetic root of the 3rd degree of $a$ is denoted as $\sqrt(a)$, the sign “$\sqrt( )$” is the sign of the arithmetic root of the 3rd degree, and the number $3$ in this notation is called root index. The number or expression that appears under the root sign is called radical.
Example 2
$\sqrt(3,5)$ – arithmetic root of the 3rd degree of $3.5$ or cube root of $3.5$;
$\sqrt(x+5)$ – arithmetic root of the 3rd degree of $x+5$ or cube root of $x+5$.
Arithmetic nth root
Definition 5
Arithmetic root nth degree from the number $a \ge 0$ a non-negative number is called which, when raised to the $n$th power, becomes equal to $a$.
Notation for the arithmetic root of degree $n$ of $a \ge 0$:
where $a$ is a radical number or expression,
