Arithmetic root examples. Square root. The Comprehensive Guide (2019)

Arithmetic root 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,

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Root degree n from a real number a, Where n- natural number, such a real number is called x, n the th power 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.

An arithmetic root of the nth degree of a non-negative number is a non-negative number nth degree which is equal to:

The power of a root is a natural number greater than 1.

3.

4.

Special cases:

1. If the root exponent is an odd integer(), then the radical expression can be negative.

In the case of an odd exponent, the equation for any real value and integer ALWAYS has a single root:

For a root of odd degree the following identity holds:

,

2. If the root exponent is an even integer (), then the radical expression cannot be negative.

In the case of an even exponent, Eq. It has

at single root

and, if and

For a root of even degree the following identity holds:

For a root of even degree the following equalities are true::

Power function, its properties and graph.

Power function and its properties.

Power function with natural exponent. The function y = x n, where n is a natural number, is called a power function with a natural exponent. For n = 1 we obtain the function y = x, its properties:

Direct proportionality. Direct proportionality is a function defined by the formula y = kx n, where the number k is called the proportionality coefficient.

Let us list the properties of the function y = kx.

The domain of a function is the set of all real numbers.

y = kx - not even function(f(- x) = k (- x)= - kx = -k(x)).

3) For k > 0 the function increases, and for k< 0 убывает на всей числовой прямой.

The graph (straight line) is shown in Figure II.1.

Rice. II.1.

When n=2 we get the function y = x 2, its properties:

Function y -x 2. Let us list the properties of the function y = x 2.

y = x 2 - even function (f(- x) = (- x) 2 = x 2 = f (x)).

The function decreases over the interval.

In fact, if , then - x 1 > - x 2 > 0, and therefore

(-x 1) 2 > (- x 2) 2, i.e., and this means the function is decreasing.

The graph of the function y=x2 is a parabola. This graph is shown in Figure II.2.

Rice. II.2.

When n = 3 we obtain the function y = x 3, its properties:

The domain of definition of a function is the entire number line.

y = x 3 - odd function (f (- x) = (- x) 2 = - x 3 = - f (x)).

3) The function y = x 3 increases along the entire number line. The graph of the function y = x 3 is shown in the figure. It is called a cubic parabola.

The graph (cubic parabola) is shown in Figure II.3.

Rice. II.3.

Let n be an arbitrary even natural number greater than two:

n = 4, 6, 8,... . In this case, the function y = x n has the same properties as the function y = x 2. The graph of such a function resembles a parabola y = x 2, only the branches of the graph at |n| >1 the steeper they go upward, the larger n, and the more “pressed” to the x axis, the larger n.

Let n be an arbitrary odd number greater than three: n = = 5, 7, 9, ... . In this case, the function y = x n has the same properties as the function y = x 3. The graph of such a function resembles a cubic parabola (only the branches of the graph go up and down the steeper, the larger n is. Note also that on the interval (0; 1) the graph of the power function y = x n moves away from the x axis more slowly as x increases, the more more than n.

Power function with negative integer exponent. Consider the function y = x - n, where n is a natural number. When n = 1 we get y = x - n or y = Properties of this function:

The graph (hyperbola) is shown in Figure II.4.

we'll decide simple task by finding the side of a square whose area is 9 cm 2. If we assume that the side of the square A cm, then we compose the equation according to the conditions of the problem:

A X A =9

A 2 =9

A 2 -9 =0

(A-3)(A+3)=0

A=3 or A=-3

The side length of a square cannot be negative number, therefore the required side of the square is 3 cm.

When solving the equation, we found the numbers 3 and -3, the squares of which are 9. Each of these numbers is called the square root of the number 9. The non-negative of these roots, that is, the number 3, is called the arithmetic root of the number.

It is quite logical to accept the fact that the root can be found from numbers to the third power (cube root), fourth power, and so on. And in principle, the root is the inverse operation of exponentiation.

Rootn th degree from the number α is such a number b, Where b n = α .

Here n- a natural number is usually called root index(or degree of root); as a rule, it is greater than or equal to 2, because the case n = 1 corny.

Designated on the letter as a symbol (root sign) on the right side is called radical. Number α - radical expression. For our example with a party, the solution could look like this: because (± 3) 2 = 9 .

We got the positive and negative values ​​of the root. This feature complicates calculations. To achieve unambiguity, the concept was introduced arithmetic root, the value of which is always with a plus sign, that is, only positive.

Root called arithmetic, if it is extracted from a positive number and is itself a positive number.

For example,

There is only one arithmetic root of a given degree from a given number.

The calculation operation is usually called “ root extraction n th degree" from among α . In essence, we perform the operation inverse to raising to a power, namely, finding the base of the power b according to a known indicator n and the result of raising to a power

α = bn.

The roots of the second and third degrees are used in practice more often than others and therefore they were given special names.

Square root: In this case, it is customary not to write the exponent 2, and the term “root” without indicating the degree most often means a square root. Geometrically interpreted, is the length of the side of a square whose area is equal to α .

Cube root: Geometrically interpreted, the length of an edge of a cube whose volume is equal to α .

Properties of arithmetic roots.

1) When calculating arithmetic root of the product, it is necessary to extract it from each factor separately

For example,

2) For calculation root of a fraction, it is necessary to extract it from the numerator and denominator of this fraction

For example,

3) When calculating root of the degree, it is necessary to divide the exponent by the root exponent

For example,

The first calculations related to extracting the square root were found in the works of mathematicians ancient Babylon and China, India, Greece (about the achievements ancient egypt there is no information in the sources in this regard).

Mathematicians of ancient Babylon (2nd millennium BC) used a special numerical method to extract the square root. The initial approximation for the square root was found based on the natural number closest to the root (in the smaller direction) n. Presenting the radical expression in the form: α=n 2 +r, we get: x 0 =n+r/2n, then an iterative refinement process was applied:

The iterations in this method converge very quickly. For ,

For example, α=5; n=2; r=1; x 0 =9/4=2.25 and we get a sequence of approximations:

In the final value, all numbers are correct except the last one.

The Greeks formulated the problem of doubling the cube, which boiled down to constructing the cube root using a compass and ruler. The rules for calculating any degree of an integer have been studied by mathematicians in India and the Arab states. Then they were widely developed in medieval Europe.

Today, for the convenience of calculating square and cube roots, calculators are widely used.