Lay down different roots. Rules for subtracting roots

The square root of a number X called a number A, which in the process of multiplying itself by itself ( A*A) can give a number X.
Those. A * A = A 2 = X, and √X = A.

Over square roots ( √x), as with other numbers, you can perform arithmetic operations such as subtraction and addition. To subtract and add roots, they must be connected using signs corresponding to these actions (for example √x- √y ).
And then bring the roots to them simplest form- if there are similar ones between them, it is necessary to make a cast. It consists in the fact that the coefficients of similar terms are taken with the signs of the corresponding terms, then they are enclosed in brackets and the common root is displayed outside the multiplier brackets. The coefficient that we have obtained is simplified according to the usual rules.

Step 1. Extracting square roots

First, to add square roots, you first need to extract these roots. This can be done if the numbers under the root sign are perfect squares. For example, take the given expression √4 + √9 . First number 4 is the square of the number 2 . Second number 9 is the square of the number 3 . Thus, the following equality can be obtained: √4 + √9 = 2 + 3 = 5 .
Everything, the example is solved. But it doesn't always happen that way.

Step 2. Taking out the multiplier of a number from under the root

If there are no full squares under the root sign, you can try to take the multiplier of the number out from under the root sign. For example, take the expression √24 + √54 .

Let's factorize the numbers:
24 = 2 * 2 * 2 * 3 ,
54 = 2 * 3 * 3 * 3 .

In list 24 we have a multiplier 4 , it can be taken out from under the square root sign. In list 54 we have a multiplier 9 .

We get the equality:
√24 + √54 = √(4 * 6) + √(9 * 6) = 2 * √6 + 3 * √6 = 5 * √6 .

Considering this example, we get the removal of the factor from under the root sign, thereby simplifying the given expression.

Step 3. Reducing the denominator

Consider the following situation: the sum of two square roots is the denominator of a fraction, for example, A / (√a + √b).
Now we are faced with the task of "getting rid of the irrationality in the denominator."
Let's use the following method: multiply the numerator and denominator of the fraction by the expression √a - √b.

We now get the abbreviated multiplication formula in the denominator:
(√a + √b) * (√a - √b) = a - b.

Similarly, if the denominator contains the difference of the roots: √a - √b, the numerator and denominator of the fraction are multiplied by the expression √a + √b.

Let's take a fraction as an example:
4 / (√3 + √5) = 4 * (√3 - √5) / ((√3 + √5) * (√3 - √5)) = 4 * (√3 - √5) / (-2) = 2 * (√5 - √3) .

An example of complex denominator reduction

Now let's consider enough complex example getting rid of irrationality in the denominator.

Let's take a fraction as an example: 12 / (√2 + √3 + √5) .
You need to take its numerator and denominator and multiply by the expression √2 + √3 - √5 .

We get:

12 / (√2 + √3 + √5) = 12 * (√2 + √3 - √5) / (2 * √6) = 2 * √3 + 3 * √2 - √30.

Step 4. Calculate the approximate value on the calculator

If you only need an approximate value, this can be done on a calculator by calculating the value of square roots. Separately, for each number, the value is calculated and recorded with the required accuracy, which is determined by the number of decimal places. Further, all the required operations are performed, as with ordinary numbers.

Estimated Calculation Example

It is necessary to calculate the approximate value of this expression √7 + √5 .

As a result, we get:

√7 + √5 ≈ 2,65 + 2,24 = 4,89 .

Please note: under no circumstances should square roots be added as prime numbers, this is completely unacceptable. That is, if we add Square root out of five and out of three, we can't get the square root of eight.

Useful advice: if you decide to factorize a number, in order to derive a square from under the root sign, you need to do a reverse check, that is, multiply all the factors that resulted from the calculations, and the final result of this mathematical calculation should be the number we were originally given.

In mathematics, roots can be square, cubic, or have any other exponent (power), which is written on the left above the root sign. The expression under the root sign is called the root expression. Root addition is similar to term addition. algebraic expression, that is, it requires the definition of similar roots.

Steps

Part 1 of 2: Finding Roots

Root designation. An expression under the root sign () means that it is necessary to extract a root of a certain degree from this expression.

• The root is denoted by a sign.
• The index (degree) of the root is written on the left above the root sign. For example, the cube root of 27 is written as: (27)
• If the exponent (degree) of the root is absent, then the exponent is considered equal to 2, that is, it is the square root (or the root of the second degree).
• The number written before the root sign is called a multiplier (that is, this number is multiplied by the root), for example 5 (2)
• If there is no factor in front of the root, then it is equal to 1 (recall that any number multiplied by 1 equals itself).
• If you are working with roots for the first time, make appropriate notes on the multiplier and exponent of the root so as not to get confused and better understand their purpose.

Remember which roots can be folded and which cannot. Just as you cannot add different terms of an expression, such as 2a + 2b 4ab, you cannot add different roots.

You cannot add roots with different root expressions, for example, (2) + (3) (5). But you can add numbers under the same root, for example, (2 + 3) = (5) (the square root of 2 is approximately 1.414, the square root of 3 is approximately 1.732, and the square root of 5 is approximately 2.236).

You cannot add roots with the same root expressions, but different exponents, for example, (64) + (64) (this sum is not equal to (64), since the square root of 64 is 8, the cube root of 64 is 4, 8 + 4 = 12, which is much larger than the fifth root of 64, which is approximately 2.297).

Part 2 of 2: Simplifying and Adding Roots

Identify and group similar roots. Similar roots are roots that have the same exponents and the same root expressions. For example, consider the expression:
2 (3) + (81) + 2 (50) + (32) + 6 (3)

• First, rewrite the expression so that roots with the same exponent are in series.
  2 (3) + 2 (50) + (32) + 6 (3) + (81)
• Then rewrite the expression so that roots with the same exponent and the same root expression are in series.
  2 (50) + (32) + 2 (3) + 6 (3) + (81)

Simplify your roots. To do this, decompose (where possible) the radical expressions into two factors, one of which is taken out from under the root. In this case, the rendered number and the root factor are multiplied.

In the example above, factor 50 into 2*25 and number 32 into 2*16. From 25 and 16, you can extract square roots (respectively 5 and 4) and take 5 and 4 out from under the root, respectively multiplying them by factors 2 and 1. Thus, you get a simplified expression: 10 (2) + 4 (2) + 2 (3) + 6 (3) + (81)

The number 81 can be factored into 3 * 27, and the cube root of 3 can be taken from the number 27. This number 3 can be taken out from under the root. Thus, you get an even more simplified expression: 10 (2) + 4 (2) + 2 (3) + 6 (3) + 3 (3)

Add the factors of similar roots. In our example, there are similar square roots of 2 (they can be added) and similar square roots of 3 (they can also be added). At cube root out of 3 there are no such roots.

10 (2) + 4 (2) = 14 (2).

2 (3)+ 6 (3) = 8 (3).

Final simplified expression: 14 (2) + 8 (3) + 3 (3)

• There are no generally accepted rules for the order in which roots are written in an expression. Therefore, you can write roots in ascending order of their exponents and in ascending order of radical expressions.

Attention, only TODAY!

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The square root of a number X called a number A, which in the process of multiplying itself by itself ( A*A) can give a number X.
Those. A * A = A 2 = X, and √X = A.

Over square roots ( √x), as with other numbers, you can perform arithmetic operations such as subtraction and addition. To subtract and add roots, they must be connected using signs corresponding to these actions (for example √x - √y ).
And then bring the roots to their simplest form - if there are similar ones between them, you need to make a cast. It consists in the fact that the coefficients of similar terms are taken with the signs of the corresponding terms, then they are enclosed in brackets and the common root is displayed outside the multiplier brackets. The coefficient that we have obtained is simplified according to the usual rules.

Step 1. Extracting square roots

First, to add square roots, you first need to extract these roots. This can be done if the numbers under the root sign are perfect squares. For example, take the given expression √4 + √9 . First number 4 is the square of the number 2 . Second number 9 is the square of the number 3 . Thus, the following equality can be obtained: √4 + √9 = 2 + 3 = 5 .
Everything, the example is solved. But it doesn't always happen that way.

Step 2. Taking out the multiplier of a number from under the root

If there are no full squares under the root sign, you can try to take the multiplier of the number out from under the root sign. For example, take the expression √24 + √54 .

Let's factorize the numbers:
24 = 2 * 2 * 2 * 3 ,
54 = 2 * 3 * 3 * 3 .

In list 24 we have a multiplier 4 , it can be taken out from under the square root sign. In list 54 we have a multiplier 9 .

We get the equality:
√24 + √54 = √(4 * 6) + √(9 * 6) = 2 * √6 + 3 * √6 = 5 * √6 .

Considering this example, we get the removal of the factor from under the root sign, thereby simplifying the given expression.

Step 3. Reducing the denominator

Consider the following situation: the sum of two square roots is the denominator of a fraction, for example, A / (√a + √b).
Now we are faced with the task of "getting rid of the irrationality in the denominator."
Let's use the following method: multiply the numerator and denominator of the fraction by the expression √a - √b.

We now get the abbreviated multiplication formula in the denominator:
(√a + √b) * (√a - √b) = a - b.

Similarly, if the denominator contains the difference of the roots: √a - √b, the numerator and denominator of the fraction are multiplied by the expression √a + √b.

Let's take a fraction as an example:
4 / (√3 + √5) = 4 * (√3 — √5) / ((√3 + √5) * (√3 — √5)) = 4 * (√3 — √5) / (-2) = 2 * (√5 — √3) .

An example of complex denominator reduction

Now we will consider a rather complicated example of getting rid of irrationality in the denominator.

Let's take a fraction as an example: 12 / (√2 + √3 + √5) .
You need to take its numerator and denominator and multiply by the expression √2 + √3 — √5 .

12 / (√2 + √3 + √5) = 12 * (√2 + √3 — √5) / (2 * √6) = 2 * √3 + 3 * √2 — √30.

Step 4. Calculate the approximate value on the calculator

If you only need an approximate value, this can be done on a calculator by calculating the value of square roots. Separately, for each number, the value is calculated and recorded with the required accuracy, which is determined by the number of decimal places. Further, all the required operations are performed, as with ordinary numbers.

Estimated Calculation Example

It is necessary to calculate the approximate value of this expression √7 + √5 .

As a result, we get:

√7 + √5 ≈ 2,65 + 2,24 = 4,89 .

Please note: under no circumstances should square roots be added as prime numbers, this is completely unacceptable. That is, if you add the square root of five and three, we cannot get the square root of eight.

Useful advice: if you decide to factorize a number, in order to derive a square from under the root sign, you need to do a reverse check, that is, multiply all the factors that resulted from the calculations, and the final result of this mathematical calculation should be the number we were originally given.

Rules for subtracting roots

1. The root of the degree from the product of non-negative numbers is equal to the product of the roots of the same degree from the factors: where (the rule for extracting the root from the product).

2. If , then y (the rule for extracting the root from a fraction).

3. If then (the rule of extracting the root from the root).

4. If then the rule for raising a root to a power).

5. If then where, i.e., the root index and the radical expression index can be multiplied by the same number.

6. If then 0, i.e., a larger positive radical expression corresponds to a larger value of the root.

7. All of the above formulas are often used in reverse order(i.e. right to left). For example,

(rule of multiplication of roots);

(the rule for dividing the roots);

8. The rule for taking the multiplier out from under the sign of the root. At

9. Inverse problem - introducing a factor under the sign of the root. For example,

10. Destruction of irrationality in the denominator of a fraction.

Let's consider some typical cases.

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For example,

11. Application of abbreviated multiplication identities to operations with arithmetic roots:

12. The factor in front of the root is called its coefficient. For example, Here 3 is a factor.

13. Roots (radicals) are called similar if they have the same root exponents and the same radical expressions, but differ only in the coefficient. To judge whether these roots (radicals) are similar or not, you need to reduce them to their simplest form.

For example, and are similar because

EXERCISES WITH SOLUTIONS

1. Simplify expressions:

Decision. 1) It makes no sense to multiply the root expression, since each of the factors represents the square of an integer. Let's use the rule of extracting the root from the product:

In the future, such actions will be performed orally.

2) Let's try, if possible, to represent the radical expression as a product of factors, each of which is the cube of an integer, and apply the rule about the root of the product:

2. Find the value of the expression:

Decision. 1) According to the rule of extracting the root from a fraction, we have:

3) We transform the radical expressions and extract the root:

3. Simplify when

Decision. When extracting a root from a root, the indices of the roots are multiplied, and the root expression remains unchanged.

If there is a coefficient before the root under the root, then before performing the operation of extracting the root, this coefficient is entered under the sign of the radical before which it stands.

Based on the above rules, we extract the last two roots:

4. Raise to a power:

Decision. When raising a root to a power, the root exponent remains unchanged, and the radical expression exponents are multiplied by the exponent.

(since it is defined, then );

If a given root has a coefficient, then this coefficient is raised to a power separately and the result is written as a coefficient at the root.

Here we used the rule that the index of the root and the index of the radical expression can be multiplied by the same number (we multiplied by i.e. divided by 2).

For example, or

4) The expression in brackets, representing the sum of two different radicals, will be cubed and simplified:

Because we have:

5. Eliminate irrationality in the denominator:

Decision. To eliminate (destroy) irrationality in the denominator of a fraction, you need to find the simplest of the expressions, which in the product with the denominator gives rational expression, and multiply the numerator and denominator of this fraction by the factor found.

For example, if there is a binomial in the denominator of a fraction, then the numerator and denominator of the fraction must be multiplied by the expression conjugate to the denominator, that is, the sum must be multiplied by the corresponding difference and vice versa.

In more difficult cases destroy irrationality not immediately, but in several steps.

1) The expression must contain

Multiplying the numerator and denominator of the fraction by we get:

2) Multiplying the numerator and denominator of the fraction by the incomplete square of the sum, we get:

3) Let's bring the fractions to a common denominator:

When solving this example, we must keep in mind that each fraction has a meaning, that is, the denominator of each fraction is different from zero. Besides,

When converting expressions containing radicals, mistakes are often made. They are caused by the inability to correctly apply the concept (definition) of the arithmetic root and the absolute value.

Rules for subtracting roots

Compute expression value

Decision.

Explanation.
To collapse the root expression, let's represent in the second factor in its root expression the number 31 as the sum of 15+16. (line 2)

After the transformation, it can be seen that the sum in the second radical expression can be represented as the square of the sum using the abbreviated multiplication formulas. (line 3)

Now let's represent each root from the given product as a degree. (line 4)

Simplify the expression (line 5)

Since the power of the product is equal to the product of the powers of each of the factors, we represent this accordingly (line 6)

As you can see, according to the formulas of abbreviated multiplication, we have the difference of the squares of two numbers. From where and calculate the value of the expression (line 7)

Calculate the value of the expression.

Decision.

Explanation.

We use the properties of the root, that the root of an arbitrary power of private numbers is equal to the private of the roots of these numbers (line 2)

The root of an arbitrary power of a number of the same degree is equal to this number (line 3)

Let's remove the minus from the bracket of the first multiplier. In this case, all characters inside the bracket will be reversed (line 4)

Let's reduce the fraction (line 5)

Let's represent the number 729 as the square of the number 27, and the number 27 as the cube of the number 3. From where we get the value of the radical expression.

Square root. First level.

Do you want to test your strength and find out the result of how ready you are for the Unified State Examination or the OGE?

1. Introduction of the concept of an arithmetic square root

The square root (arithmetic square root) of a non-negative number is a non-negative number whose square is equal.
.

The number or expression under the root sign must be non-negative

2. Table of squares

3. Properties of the arithmetic square root

Introduction to the concept of arithmetic square root

Let's try to figure out what kind of concept a "root" is and "what it is eaten with." To do this, consider examples that you have already encountered in the lessons (well, or you just have to face this).

For example, we have an equation. What is the solution given equation? What numbers can be squared and get at the same time? Remembering the multiplication table, you can easily give the answer: and (because when you multiply two negative numbers, you get a positive number)! To simplify, mathematicians have introduced a special concept of the square root and assigned it a special symbol.

Let's define the arithmetic square root.

Why does the number have to be non-negative? For example, what is equal to? Okay, let's try to figure it out. Maybe three? Let's check: and not. Maybe, ? Again, check: Well, is it not selected? This is to be expected - because there are no numbers that, when squared, give a negative number!

However, you probably already noticed that the definition says that the solution of the square root of "a number is a non-negative number whose square is equal to". And at the very beginning, we analyzed the example, selected numbers that can be squared and obtained at the same time, the answer was and, and here it is talking about some kind of “non-negative number”! Such a remark is quite appropriate. Here it is necessary simply to distinguish between the concepts of quadratic equations and the arithmetic square root of a number. For example, it is not equivalent to an expression.

And it follows that.

Of course, this is very confusing, but it must be remembered that the signs are the result of solving the equation, since when solving the equation, we must write down all the x's that, when substituted into the original equation, will give the correct result. In our quadratic equation fits both and.

However, if you just take the square root of something, then you always get one non-negative result.

Now try to solve this equation. Everything is not so simple and smooth, right? Try to sort through the numbers, maybe something will burn out?

Let's start from the very beginning - from scratch: - does not fit, move on; - less than three, we also brush aside, but what if? Let's check: - also does not fit, because it's more than three. With negative numbers, the same story will turn out. And what to do now? Did the search 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, it is obvious that the solutions will not be integers. Moreover, they are not rational. So, what is next? Let's build a graph of the function and mark the solutions on it.

Let's try to trick the system and get an answer using a calculator! Let's get the root out of business! Oh-oh-oh, it turns out that Such a number never ends. How can you remember this, because there will be no calculator on the exam !? Everything is very simple, you don’t need to remember it, you need to remember (or be able to quickly estimate) an approximate value. and the answers themselves. Such numbers are called irrational, and it was to simplify the notation of such numbers that the concept of a square root was introduced.
Let's look at another example to reinforce. Let's analyze the following problem: you need to cross diagonally a square field with a side of km, how many km do you have to go?

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.

Root extraction

So that solving examples with roots does not cause problems, you need to see and recognize them. To do this, you need to know at least the squares of numbers from to, as well as be able to recognize them.

That is, you need to know what is squared, and also, conversely, what is squared. At first, this table will help you in extracting the root.

As soon as you solve a sufficient number of examples, then the need for it will automatically disappear.
Try to extract the square root in the following expressions yourself:

Well, how did it work? Now let's see these examples:

Properties of the arithmetic square root

Now you know how to extract roots and it's time to learn about the properties of the arithmetic square root. There are only 3 of them:

• multiplication;
• division;
• exponentiation.

Well, they are just very easy to remember with the help of this table and, of course, training:

How to decide
quadratic equations

In the previous lessons, we analyzed "How to solve linear equations", that is, equations of the first degree. In this lesson, we will explore what is a quadratic equation and how to solve it.

What is a quadratic equation

The degree of an equation is determined by the highest degree to which the unknown stands.

If the maximum degree to which the unknown stands is “2”, then you have a quadratic equation.

Examples of quadratic equations

• 5x2 - 14x + 17 = 0
• −x 2 + x +

To find "a", "b" and "c" you need to compare your equation with the general form of the quadratic equation "ax 2 + bx + c = 0".

Let's practice determining the coefficients "a", "b" and "c" in quadratic equations.

• a=5
• b = −14
• c = 17

• a = −7
• b = −13
• c = 8

• a = −1
• b = 1

• a = 1
• b = 0.25
• c = 0

• a = 1
• b = 0
• c = −8

How to solve quadratic equations

Unlike linear equations to solve quadratic equations, a special formula for finding roots.

To solve a quadratic equation you need:

• bring the quadratic equation to general view" ax 2 + bx + c = 0 ". That is, only "0" should remain on the right side;
• use the formula for roots:

Let's use an example to figure out how to apply the formula to find the roots of a quadratic equation. Let's solve the quadratic equation.

The equation "x 2 − 3x − 4 = 0" has already been reduced to the general form "ax 2 + bx + c = 0" and does not require additional simplifications. To solve it, we need only apply formula for finding the roots of a quadratic equation.

Let's define coefficients "a", "b" and "c" for this equation.

• a = 1
• b = −3
• c = −4

Substitute them in the formula and find the roots.

Be sure to memorize the formula for finding roots.

With its help, any quadratic equation is solved.

Consider another example of a quadratic equation.

In this form, it is quite difficult to determine the coefficients "a", "b" and "c". Let's first bring the equation to the general form "ax 2 + bx + c = 0".

Now you can use the formula for the roots.

There are times when there are no roots in quadratic equations. This situation occurs when a negative number appears in the formula under the root.

We remember from the definition of the square root that you cannot take the square root of a negative number.

Consider an example of a quadratic equation that has no roots.

So, we got a situation where there is a negative number under the root. This means that there are no roots in the equation. Therefore, in response, we wrote down "There are no real roots."

What do the words "no real roots" mean? Why can't you just write "no roots"?

In fact, there are roots in such cases, but within the framework of school curriculum they are not passed, therefore, in response, we write down that among real numbers there are no roots. In other words, "There are no real roots."

Incomplete quadratic equations

Sometimes there are quadratic equations in which there are no explicit coefficients "b" and/or "c". For example, in this equation:

Such equations are called incomplete. quadratic equations. How to solve them is discussed in the lesson "Incomplete Quadratic Equations".

Extracting the square root of a number is not the only operation that can be performed with this mathematical phenomenon. Just like ordinary numbers, square roots can be added and subtracted.

Yandex.RTB R-A-339285-1

Rules for adding and subtracting square roots

Definition 1

Actions such as adding and subtracting a square root are possible only if the root expression is the same.

Example 1

You can add or subtract expressions 2 3 and 6 3, but not 5 6 and 9 4 . If it is possible to simplify the expression and bring it to roots with the same root number, then simplify, and then add or subtract.

Root Actions: The Basics

Example 2

6 50 - 2 8 + 5 12

Action algorithm:

1. Simplify the root expression. To do this, it is necessary to decompose the root expression into 2 factors, one of which is a square number (the number from which the whole square root is extracted, for example, 25 or 9).
2. Then you need to extract the root from square number and write the resulting value before the root sign. Please note that the second factor is entered under the root sign.
3. After the simplification process, it is necessary to underline the roots with the same radical expressions - only they can be added and subtracted.
4. For roots with the same radical expressions, it is necessary to add or subtract the factors that precede the root sign. The root expression remains unchanged. Do not add or subtract root numbers!

Tip 1

If you have an example with large quantity identical radical expressions, then underline such expressions with single, double and triple lines to facilitate the calculation process.

Example 3

Let's try this example:

6 50 = 6 (25 × 2) = (6 × 5) 2 = 30 2 . First you need to decompose 50 into 2 factors 25 and 2, then take the root of 25, which is 5, and take 5 out from under the root. After that, you need to multiply 5 by 6 (the multiplier at the root) and get 30 2 .

2 8 = 2 (4 × 2) = (2 × 2) 2 = 4 2 . First, you need to decompose 8 into 2 factors: 4 and 2. Then, from 4, extract the root, which is equal to 2, and take 2 out from under the root. After that, you need to multiply 2 by 2 (the factor at the root) and get 4 2 .

5 12 = 5 (4 × 3) = (5 × 2) 3 = 10 3 . First, you need to decompose 12 into 2 factors: 4 and 3. Then extract the root from 4, which is 2, and take it out from under the root. After that, you need to multiply 2 by 5 (the factor at the root) and get 10 3 .

Simplification result: 30 2 - 4 2 + 10 3

30 2 - 4 2 + 10 3 = (30 - 4) 2 + 10 3 = 26 2 + 10 3 .

As a result, we saw how many identical radical expressions are contained in this example. Now let's practice with other examples.

Example 4

• Simplify (45) . We factorize 45: (45) = (9 × 5) ;
• We take out 3 from under the root (9 \u003d 3): 45 \u003d 3 5;
• We add the factors at the roots: 3 5 + 4 5 = 7 5 .

Example 5

6 40 - 3 10 + 5:

• Simplifying 6 40 . We factorize 40: 6 40 \u003d 6 (4 × 10) ;
• We take out 2 from under the root (4 \u003d 2): 6 40 \u003d 6 (4 × 10) \u003d (6 × 2) 10;
• We multiply the factors that are in front of the root: 12 10;
• We write the expression in a simplified form: 12 10 - 3 10 + 5;
• Since the first two terms have the same root numbers, we can subtract them: (12 - 3) 10 = 9 10 + 5.

Example 6

As we can see, it is not possible to simplify the radical numbers, so we look for members with the same radical numbers in the example, perform mathematical operations (add, subtract, etc.) and write the result:

(9 - 4) 5 - 2 3 = 5 5 - 2 3 .

Adviсe:

• Before adding or subtracting, it is imperative to simplify (if possible) the radical expressions.
• Adding and subtracting roots with different root expressions is strictly prohibited.
• Do not add or subtract an integer or square root: 3 + (2 x) 1 / 2 .
• When performing actions with fractions, you need to find a number that is completely divisible by each denominator, then bring the fractions to a common denominator, then add the numerators, and leave the denominators unchanged.

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