Root formulas. Properties of square roots.

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In the previous lesson we figured out what a square root is. It's time to figure out which ones exist formulas for roots what are properties of roots, and what can be done with all this.

Formulas of roots, properties of roots and rules for working with roots- this is essentially the same thing. Formulas for square roots surprisingly little. Which certainly makes me happy! Or rather, you can write a lot of different formulas, but for practical and confident work with roots, only three are enough. Everything else flows from these three. Although many people get confused in the three root formulas, yes...

Let's start with the simplest one. Here she is:

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You can get acquainted with functions and derivatives.

It's time to sort it out root extraction methods. They are based on the properties of roots, in particular, on equality, which is true for any Not negative number b.

Below we will look at the main methods of extracting roots one by one.

Let's start with the simplest case - extracting roots from natural numbers using a table of squares, a table of cubes, etc.

If tables of squares, cubes, etc. If you don’t have it at hand, it’s logical to use the method of extracting the root, which involves decomposing the radical number into prime factors.

It is worth special mentioning what is possible for roots with odd exponents.

Finally, let's consider a method that allows us to sequentially find the digits of the root value.

Let's get started.

Using a table of squares, a table of cubes, etc.

In the simplest cases, tables of squares, cubes, etc. allow you to extract roots. What are these tables?

The table of squares of integers from 0 to 99 inclusive (shown below) consists of two zones. The first zone of the table is located on a gray background; by selecting a specific row and a specific column, it allows you to compose a number from 0 to 99. For example, let’s select a row of 8 tens and a column of 3 units, with this we fixed the number 83. The second zone occupies the rest of the table. Each cell is located at the intersection of a certain row and a certain column, and contains the square of the corresponding number from 0 to 99. At the intersection of our chosen row of 8 tens and column 3 of ones there is a cell with the number 6,889, which is the square of the number 83.

Tables of cubes, tables of fourth powers of numbers from 0 to 99, and so on are similar to the table of squares, only they contain cubes, fourth powers, etc. in the second zone. corresponding numbers.

Tables of squares, cubes, fourth powers, etc. allow you to extract square roots, cube roots, fourth roots, etc. accordingly from the numbers in these tables. Let us explain the principle of their use when extracting roots.

Let's say we need to extract the nth root of the number a, while the number a is contained in the table of nth powers. Using this table we find the number b such that a=b n. Then , therefore, the number b will be the desired root of the nth degree.

As an example, let's show how to use a cube table to extract the cube root of 19,683. We find the number 19,683 in the table of cubes, from it we find that this number is the cube of the number 27, therefore, .

It is clear that tables of nth powers are very convenient for extracting roots. However, they are often not at hand, and compiling them requires some time. Moreover, it is often necessary to extract roots from numbers that are not contained in the corresponding tables. In these cases, you have to resort to other methods of root extraction.

Factoring a radical number into prime factors

A fairly convenient way to extract the root of a natural number (if, of course, the root is extracted) is to decompose the radical number into prime factors. His the point is this: after that it is quite easy to represent it as a power with the desired exponent, which allows you to obtain the value of the root. Let's clarify this point.

Let the nth root of a natural number a be taken and its value equal b. In this case, the equality a=b n is true. Number b like any natural number can be represented as the product of all its prime factors p 1 , p 2 , …, p m in the form p 1 · p 2 · … · p m , and the radical number a in this case is represented as (p 1 · p 2 · … · p m) n. Since the decomposition of a number into prime factors is unique, the decomposition of the radical number a into prime factors will have the form (p 1 ·p 2 ·…·p m) n, which makes it possible to calculate the value of the root as.

Note that if the decomposition into prime factors of a radical number a cannot be represented in the form (p 1 ·p 2 ·…·p m) n, then the nth root of such a number a is not completely extracted.

Let's figure this out when solving examples.

Example.

Take the square root of 144.

Solution.

If you look at the table of squares given in the previous paragraph, you can clearly see that 144 = 12 2, from which it is clear that the square root of 144 is equal to 12.

But in light of this point, we are interested in how the root is extracted by decomposing the radical number 144 into prime factors. Let's look at this solution.

Let's decompose 144 to prime factors:

That is, 144=2·2·2·2·3·3. Based on the resulting decomposition, the following transformations can be carried out: 144=2·2·2·2·3·3=(2·2) 2·3 2 =(2·2·3) 2 =12 2. Hence, .

Using the properties of the degree and the properties of the roots, the solution could be formulated a little differently: .

Answer:

To consolidate the material, consider the solutions to two more examples.

Example.

Calculate the value of the root.

Solution.

The prime factorization of the radical number 243 has the form 243=3 5 . Thus, .

Answer:

Example.

Is the root value an integer?

Solution.

To answer this question, let's factor the radical number into prime factors and see if it can be represented as a cube of an integer.

We have 285 768=2 3 ·3 6 ·7 2. The resulting expansion cannot be represented as a cube of an integer, since the power of the prime factor 7 is not a multiple of three. Therefore, the cube root of 285,768 cannot be extracted completely.

Answer:

No.

Extracting roots from fractional numbers

It's time to figure out how to extract the root from fractional number. Let the fractional radical number be written as p/q. According to the property of the root of a quotient, the following equality is true. From this equality it follows rule for extracting the root of a fraction: The root of a fraction is equal to the quotient of the root of the numerator divided by the root of the denominator.

Let's look at an example of extracting a root from a fraction.

Example.

What is the square root of common fraction 25/169 .

Solution.

Using the table of squares, we find that the square root of the numerator of the original fraction is equal to 5, and the square root of the denominator is equal to 13. Then . This completes the extraction of the root of the common fraction 25/169.

Answer:

The root of a decimal fraction or mixed number is extracted after replacing the radical numbers with ordinary fractions.

Example.

Take the cube root of the decimal fraction 474.552.

Solution.

Let's imagine the original decimal as a common fraction: 474.552=474552/1000. Then . It remains to extract the cube roots that are in the numerator and denominator of the resulting fraction. Because 474 552=2·2·2·3·3·3·13·13·13=(2 3 13) 3 =78 3 and 1 000 = 10 3, then And . All that remains is to complete the calculations .

Answer:

.

Taking the root of a negative number

It is worthwhile to dwell on extracting roots from negative numbers. When studying roots, we said that when the root exponent is an odd number, then there can be a negative number under the root sign. We gave these entries the following meaning: for a negative number −a and an odd exponent of the root 2 n−1, . This equality gives rule for extracting odd roots from negative numbers: to extract the root of a negative number, you need to take the root of the opposite positive number, and put a minus sign in front of the result.

Let's look at the example solution.

Example.

Find the value of the root.

Solution.

Let's transform the original expression so that there is a positive number under the root sign: . Now mixed number replace it with an ordinary fraction: . We apply the rule for extracting the root of an ordinary fraction: . It remains to calculate the roots in the numerator and denominator of the resulting fraction: .

Here is a short summary of the solution: .

Answer:

.

Bitwise determination of the root value

IN general case under the root there is a number that, using the techniques discussed above, cannot be represented as the nth power of any number. But at the same time there is a need to know the meaning given root, at least up to a certain sign. In this case, to extract the root, you can use an algorithm that allows you to sequentially obtain a sufficient number of digit values ​​of the desired number.

The first step of this algorithm is to find out what the most significant bit of the root value is. To do this, the numbers 0, 10, 100, ... are sequentially raised to the power n until the moment when a number exceeds the radical number is obtained. Then the number that we raised to the power n at the previous stage will indicate the corresponding most significant digit.

For example, consider this step of the algorithm when extracting the square root of five. Take the numbers 0, 10, 100, ... and square them until we get a number greater than 5. We have 0 2 =0<5 , 10 2 =100>5, which means the most significant digit will be the ones digit. The value of this bit, as well as the lower ones, will be found in the next steps of the root extraction algorithm.

All subsequent steps of the algorithm are aimed at sequentially clarifying the value of the root by finding the values ​​of the next bits of the desired value of the root, starting with the highest one and moving to the lowest ones. For example, the value of the root at the first step turns out to be 2, at the second – 2.2, at the third – 2.23, and so on 2.236067977…. Let us describe how the values ​​of the digits are found.

The digits are found by searching through their possible values ​​0, 1, 2, ..., 9. In this case, the nth powers of the corresponding numbers are calculated in parallel, and they are compared with the radical number. If at some stage the value of the degree exceeds the radical number, then the value of the digit corresponding to the previous value is considered found, and the transition to the next step of the root extraction algorithm is made; if this does not happen, then the value of this digit is 9.

Let us explain these points using the same example of extracting the square root of five.

First we find the value of the units digit. We will go through the values ​​0, 1, 2, ..., 9, calculating 0 2, 1 2, ..., 9 2, respectively, until we get a value greater than the radical number 5. It is convenient to present all these calculations in the form of a table:

So the value of the units digit is 2 (since 2 2<5 , а 2 3 >5 ). Let's move on to finding the value of the tenths place. In this case, we will square the numbers 2.0, 2.1, 2.2, ..., 2.9, comparing the resulting values ​​with the radical number 5:

Since 2.2 2<5 , а 2,3 2 >5, then the value of the tenths place is 2. You can proceed to finding the value of the hundredths place:

This is how the next value of the root of five was found, it is equal to 2.23. And so you can continue to find values: 2,236, 2,2360, 2,23606, 2,236067, … .

To consolidate the material, we will analyze the extraction of the root with an accuracy of hundredths using the considered algorithm.

First we determine the most significant digit. To do this, we cube the numbers 0, 10, 100, etc. until we get a number greater than 2,151,186. We have 0 3 =0<2 151,186 , 10 3 =1 000<2151,186 , 100 3 =1 000 000>2 151,186 , so the most significant digit is the tens digit.

Let's determine its value.

Since 10 3<2 151,186 , а 20 3 >2 151.186, then the value of the tens place is 1. Let's move on to units.

Thus, the value of the ones digit is 2. Let's move on to tenths.

Since even 12.9 3 is less than the radical number 2 151.186, then the value of the tenths place is 9. It remains to perform the last step of the algorithm; it will give us the value of the root with the required accuracy.

At this stage, the value of the root is found accurate to hundredths: .

In conclusion of this article, I would like to say that there are many other ways to extract roots. But for most tasks, the ones we studied above are sufficient.

Bibliography.

• Makarychev Yu.N., Mindyuk N.G., Neshkov K.I., Suvorova S.B. Algebra: textbook for 8th grade. educational institutions.
• Kolmogorov A.N., Abramov A.M., Dudnitsyn Yu.P. and others. Algebra and the beginnings of analysis: Textbook for grades 10 - 11 of general education institutions.
• Gusev V.A., Mordkovich A.G. Mathematics (a manual for those entering technical schools).

A radical expression is an algebraic expression that is under the sign of a root (square, cubic, or higher order). Sometimes the meanings of different expressions can be the same, for example, 1/(√2 - 1) = √2 + 1. Simplification of the radical expression is intended to bring it to some canonical form of notation. If two expressions that are written in canonical form are still different, their values ​​are not equal. In mathematics, it is believed that the canonical form of writing radical expressions (as well as expressions with roots) corresponds to the following rules:

• If possible, get rid of the fraction under the root sign
• Get rid of expressions with fractional exponents
• If possible, get rid of the roots in the denominator
• Get rid of the root-by-root multiplication operation
• Under the root sign, you need to leave only those terms from which it is impossible to extract an integer root

These rules can be applied to test tasks. For example, if you solved a problem, but the result does not match any of the answers given, write the result in canonical form. Keep in mind that answers to test tasks are given in canonical form, so if you write the result in the same form, you can easily determine the correct answer. If a problem requires “simplifying the answer” or “simplifying radical expressions,” it is necessary to write the result in canonical form. Moreover, the canonical form makes solving equations easier, although some equations are easier to solve if you forget about the canonical notation for a while.

Steps

Getting rid of full squares and full cubes

Getting rid of an expression with a fractional exponent

Convert the expression with a fractional exponent into a radical expression. Or, if necessary, convert the radical expression into a fractional expression, but never mix such expressions in one equation, for example, like this: √5 + 5^(3/2). Let's say you decide to work with the roots; We will denote the square root of n as √n, and the cubic root of n as the cube√n.

Getting rid of fractions under the root sign

According to the canonical form of notation, the root of a fraction must be represented as a division of the roots of integers.

Look at the radical expression. If it is a fraction, go to the next step.

Replace the root of the fraction with the ratio of the two roots according to the following identity:√(a/b) = √a/√b.

• Do not use this identity if the denominator is negative or includes a variable that may be negative. In this case, simplify the fraction first.

1. Simplify the perfect squares (if you have them). For example, √(5/4) = √5/√4 = (√5)/2.

Eliminating the operation of multiplying roots

Getting rid of factors that are perfect squares

Lay out radical number into factors. Factors are some numbers that, when multiplied, produce the original number. For example, 5 and 4 are two factors of the number 20. If an integer root cannot be extracted from a radical number, factor the number into its possible factors and find a perfect square among them.

• For example, write down all the factors of 45: 1, 3, 5, 9, 15, 45. 9 is a factor of 45 (9 x 5 = 45) and a perfect square (9 = 3^2).

1. Take the multiplier, which is a perfect square, beyond the root sign. 9 is a perfect square because 3 x 3 = 9. Get rid of the 9 under the root sign and write a 3 before the root sign; under the root sign there will be 5. If you put the number 3 under the root sign, it will be multiplied by itself and by the number 5, that is, 3 x 3 x 5 = 9 x 5 = 45. Thus, 3√ 5 is a simplified form of notation √45.

   • √45 = √(9 * 5) = √9 * √5 = 3√5.

2. Find the perfect square in the radical expression with the variable. Remember: √(a^2) = |a|. Such an expression can be simplified to "a", but only if the variable takes positive values. √(a^3) can be decomposed into √a * √(a^2), because when identical variables are multiplied, their exponents add up (a * a^2 = a^3).

   • Thus, in the expression a^3, the perfect square is a^2.

3. Take out the variable that is a perfect square outside the root sign. Get rid of the a^2 under the root sign and write an "a" before the root sign. Thus, √(a^3) = a√a.

   Give similar terms and simplify any rational expressions.

Getting rid of roots in the denominator (rationalization of the denominator)

According to the canonical form denominator, if possible, should include only integers (or a polynomial if a variable is present).

• If the denominator is a radical monomial, such as [numerator]/√5, multiply the numerator and denominator by that root: ([numerator] * √5)/(√5 * √5) = ([numerator] * √5 )/5.
  • For a cube root or greater root, multiply the numerator and denominator by the root with the radical to the appropriate power to rationalize the denominator. If, for example, the denominator is the cube of √5, multiply the numerator and denominator by the cube of √(5^2).
• If the denominator is a sum or difference of square roots, such as √2 + √6, multiply the numerator and denominator by the conjugate, that is, the expression with the opposite sign between its terms. For example: [numerator]/(√2 + √6) = ([numerator] * (√2 - √6))/((√2 + √6) * (√2 - √6)). Then use the difference of squares formula ((a + b)(a - b) = a^2 - b^2) to rationalize the denominator: (√2 + √6)(√2 - √6) = (√2)^2 - (√6)^2 = 2 - 6 = -4.
  • The difference of squares formula can also be applied to an expression of the form 5 + √3 because any integer is the square root of another integer. For example: 1/(5 + √3) = (5 - √3)/((5 + √3)(5 - √3)) = (5 - √3)/(5^2 - (√3)^ 2) = (5 - √3)/(25 - 3) = (5 - √3)/22
  • This method can be applied to the sum of square roots such as √5 - √6 + √7. If you group this expression in the form (√5 - √6) + √7 and multiply it by (√5 - √6) - √7, you will not get rid of the roots, but will get an expression of the form a + b * √30, where " a" and "b" are monomials without a root. Then the resulting expression can be multiplied by its conjugate: (a + b * √30)(a - b * √30) to get rid of the roots. That is, if a conjugate expression can be used once to get rid of a certain number of roots, then it can be used as many times as necessary to get rid of all the roots.
  • This method also applies to roots of higher powers, such as the expression "4th root of 3 plus 7th root of 9." In this case, multiply the numerator and denominator by the conjugate expression of the denominator. But here the conjugate expression will be slightly different compared to those described above. You can read about this case in algebra textbooks.

1. Simplify the numerator after you have removed the roots in the denominator. The numerator is the product of the original expression and the conjugate expression.

When solving some mathematical problems, you have to operate with square roots. Therefore, it is important to know the rules of operations with square roots and learn how to transform expressions containing them. The goal is to study the rules of operations with square roots and ways to transform expressions with square roots.

We know that some rational numbers are expressed as infinite periodic decimal fractions, such as the number 1/1998=0.000500500500... But nothing prevents us from imagining a number whose decimal expansion does not reveal any period. Such numbers are called irrational.

The history of irrational numbers dates back to the amazing discovery of the Pythagoreans back in the 6th century. BC e. It all started with a seemingly simple question: what number expresses the length of the diagonal of a square with side 1?

The diagonal divides the square into 2 identical right-angled triangles, in each of which it acts as a hypotenuse. Therefore, as follows from the Pythagorean theorem, the length of the diagonal of a square is equal to

. The temptation immediately arises to take out a microcalculator and press the square root key. On the scoreboard we will see 1.4142135. A more advanced calculator that performs calculations with high accuracy will show 1.414213562373. And with the help of a modern powerful computer you can calculate with an accuracy of hundreds, thousands, millions of decimal places. But even the most powerful computer, no matter how long it runs, will never be able to calculate all the decimal digits or detect any period in them.

And although Pythagoras and his students did not have a computer, they were the ones who substantiated this fact. The Pythagoreans proved that the diagonal of a square and its side have no common measure (i.e., a segment that would be plotted an integer number of times both on the diagonal and on the side). Therefore, the ratio of their lengths is the number

– cannot be expressed as the ratio of some integers m and n. And since this is so, we add, the decimal expansion of a number does not reveal any regular pattern.

Following the discovery of the Pythagoreans

How to prove that a number

irrational? Suppose there is a rational number m/n=. We will consider the fraction m/n irreducible, because a reducible fraction can always be reduced to an irreducible one. Raising both sides of the equality, we get . From here we conclude that m is an even number, that is, m = 2K. Therefore and, therefore, , or . But then we get that n is an even number, but this cannot be, since the fraction m/n is irreducible. A contradiction arises.

It remains to conclude that our assumption is incorrect and the rational number m/n is equal to

does not exist.

1. Square root of a number

Knowing the time t , you can find the path in free fall using the formula:

Let's solve the inverse problem.

Task . How many seconds will it take for a stone dropped from a height of 122.5 m to fall?

To find the answer, you need to solve the equation

From it we find that Now it remains to find a positive number t such that its square is 25. This number is 5, since So the stone will fall for 5 s.

You also have to look for a positive number by its square when solving other problems, for example, when finding the length of the side of a square by its area. Let us introduce the following definition.

Definition . A non-negative number whose square is equal to a non-negative number a is called the square root of a. This number stands for

Thus

Example . Because

You cannot take square roots from negative numbers, since the square of any number is either positive or equal to zero. For example, the expression

has no numerical value. the sign is called the radical sign (from the Latin “radix” - root), and the number A- radical number. For example, in the notation the radical number is 25. Since This means that the square root of the number written by one and 2n zeros, is equal to the number written by one and n zeros: = 10…0

2n zeros n zeros

Similarly, it is proved that

2n zeros n zeros

For example,

2. Calculating square roots

We know that there is no rational number whose square is 2. This means that

cannot be a rational number. It is an irrational number, i.e. is written as a non-periodic infinite decimal fraction, and the first decimal places of this fraction are 1.414... To find the next decimal place, you need to take the number 1.414 X, Where X can take the values ​​0, 1, 2, 3, 4, 5, 6, 7, 8, 9, square these numbers in order and find such a value X, in which the square is less than 2, but the next square is greater than 2. This value is x=2. Next, we repeat the same thing with numbers like 1.4142 X. Continuing this process, we obtain one after another the digits of the infinite decimal fraction equal to .

The existence of a square root of any positive real number is proved in a similar way. Of course, sequential squaring is a very labor-intensive task, and therefore there are ways to quickly find the decimal places of the square root. Using a microcalculator you can find the value

with eight correct numbers. To do this, just enter the number into the microcalculator a>0 and press the key - 8 digits of the value will be displayed on the screen. In some cases it is necessary to use the properties of square roots, which we will indicate below.

If the accuracy provided by a microcalculator is insufficient, you can use the method for refining the value of the root given by the following theorem.

Theorem. If a is a positive number and is an approximate value for by excess, then

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