addition of two roots. Rules for adding square roots

Addition and subtraction of roots- one of the most common "stumbling blocks" for those who take a course in mathematics (algebra) in high school. However, learning how to add and subtract them correctly is very important, because examples for the sum or difference of roots are included in the program of the basic Unified State Exam in the discipline "mathematics".

In order to master the solution of such examples, you need two things - to understand the rules, as well as to gain practice. Having solved one or two dozen typical examples, the student will bring this skill to automatism, and then he will have nothing to fear at the exam. It is recommended to start mastering arithmetic operations with addition, because adding them is a little easier than subtracting them.

The easiest way to explain this is with the example of a square root. In mathematics, there is a well-established term "square". "Square" means to multiply a specific number by itself once.. For example, if you square 2, you get 4. If you square 7, you get 49. The square of 9 is 81. So the square root of 4 is 2, of 49 is 7, and of 81 is 9.

As a rule, teaching this topic in mathematics begins with square roots. In order to immediately determine it, a high school student must know the multiplication table by heart. For those who do not know this table well, you have to use hints. Usually, the process of extracting the root square from a number is given in the form of a table on the covers of many school mathematics notebooks.

Roots are of the following types:

• square;
• cubic (or the so-called third degree);
• fourth degree;
• fifth degree.

Addition rules

In order to successfully solve a typical example, it must be borne in mind that not all root numbers can be stacked with each other. In order to be able to put them together, they must be brought to a single pattern. If this is not possible, then the problem has no solution. Such problems are also often found in mathematics textbooks as a kind of trap for students.

Addition is not allowed in assignments when the radical expressions differ from each other. This can be illustrated with an illustrative example:

• the student is faced with the task: to add the square root of 4 and of 9;
• an inexperienced student who does not know the rule usually writes: "root of 4 + root of 9 \u003d root of 13."
• it is very easy to prove that this way of solving is wrong. To do this, you need to find the square root of 13 and check if the example is solved correctly;
• using a microcalculator, you can determine that it is approximately 3.6. Now it remains to check the solution;
• root of 4=2, and of 9=3;
• The sum of two and three is five. Thus, this solution algorithm can be considered incorrect.

If the roots have the same degree, but different numerical expressions, it is taken out of brackets, and the sum of two radical expressions. Thus, it is already extracted from this amount.

Addition algorithm

In order to correctly solve the simplest problem, it is necessary:

1. Determine what exactly requires addition.
2. Find out if it is possible to add values ​​​​to each other, guided by the rules existing in mathematics.
3. If they cannot be added, you need to transform them in such a way that they can be added.
4. Having carried out all the necessary transformations, it is necessary to perform addition and write down the finished answer. Addition can be done mentally or with a calculator, depending on the complexity of the example.

What are similar roots

In order to correctly solve an addition example, it is necessary, first of all, to think about how it can be simplified. To do this, you need to have a basic knowledge of what similarity is.

The ability to identify similar ones helps to quickly solve the same type of addition examples, bringing them into a simplified form. To simplify a typical addition example, you need to:

1. Find similar ones and allocate them to one group (or several groups).
2. Rewrite the existing example in such a way that the roots that have the same indicator follow each other clearly (this is called "grouping").
3. Next, you should write the expression again again, this time in such a way that similar ones (which have the same indicator and the same root figure) also follow each other.

After that, a simplified example is usually easy to solve.

In order to correctly solve any addition example, you need to clearly understand the basic rules of addition, and also know what a root is and how it happens.

Sometimes such tasks look very complicated at first glance, but usually they are easily solved by grouping similar ones. The most important thing is practice, and then the student will begin to "click tasks like nuts." Root addition is one of the most important branches of mathematics, so teachers should allocate enough time to study it.

Video

This video will help you understand the equations with square roots.

In mathematics, any action has its own pair-opposite - in essence, this is one of the manifestations of the Hegelian law of dialectics: "the unity and struggle of opposites." One of the actions in such a “pair” is aimed at increasing the number, and the other, the opposite of it, is decreasing. For example, the action opposite to addition is subtraction, and division corresponds to multiplication. Raising to a power also has its own dialectical pair-opposite. It's about root extraction.

To extract the root of such and such a degree from a number means to calculate which number must be raised to the corresponding power in order to end up with this number. The two degrees have their own separate names: the second degree is called the "square", and the third - the "cube". Accordingly, it is pleasant to call the roots of these degrees the square root and the cubic root. Actions with cube roots are a topic for a separate discussion, but now let's talk about adding square roots.

Let's start with the fact that in some cases it is easier to extract square roots first, and then add the results. Suppose we need to find the value of such an expression:

After all, it is not at all difficult to calculate that the square root of 16 is 4, and of 121 - 11. Therefore,

√16+√121=4+11=15

However, this is the simplest case - here we are talking about full squares, i.e. about numbers that are obtained by squaring whole numbers. But this is not always the case. For example, the number 24 is not a perfect square (there is no such integer that, when raised to the second power, would result in 24). The same applies to a number like 54 ... What if we need to add the square roots of these numbers?

In this case, we will get in the answer not a number, but another expression. The maximum that we can do here is to simplify the original expression as much as possible. To do this, you will have to take out the factors from under the square root. Let's see how this is done using the mentioned numbers as an example:

To begin with, we factorize 24 - in such a way that one of them can easily be taken as a square root (i.e., so that it is a perfect square). There is such a number - this is 4:

Now let's do the same with 54. In its composition, this number will be 9:

Thus, we get the following:

√24+√54=√(4*6)+ √(9*6)

Now let's extract the roots from what we can extract them from: 2*√6+3*√6

There is a common factor here, which we can take out of brackets:

(2+3)* √6=5*√6

This will be the result of the addition - nothing else can be extracted here.

True, you can resort to the help of a calculator - however, the result will be approximate and with a huge number of decimal places:

√6=2,449489742783178

Gradually rounding it up, we get approximately 2.5. If we still would like to bring the solution of the previous example to its logical conclusion, we can multiply this result by 5 - and we get 12.5. A more accurate result with such initial data cannot be obtained.

The square root of the number x is the number a, which, when multiplied by itself, gives the number x: a * a = a^2 = x, ?x = a. As with any numbers, it is allowed to perform the arithmetic operations of addition and subtraction over square roots.

Instruction

1. First, when adding square roots, try to extract those roots. This will be valid if the numbers under the root sign are perfect squares. Let's say the expression?4 +?9 is given. The first number 4 is the square of the number 2. The second number 9 is the square of the number 3. So it turns out that: ?4 + ?9 = 2 + 3 = 5.

2. If there are no full squares under the root sign, then try to transfer the multiplier of the number from under the root sign. Let's say, let the expression? 24 +? 54 be given. Factorize the numbers: 24 = 2 * 2 * 2 * 3, 54 = 2 * 3 * 3 * 3. In the number 24 there is a factor 4, the one that can be transferred from the square root sign. In the number 54, there is a factor of 9. Thus, it turns out that: In this example, as a result of removing the factor from the root sign, it turned out to simplify the given expression.

3. Let the sum of 2 square roots be the denominator of a fraction, say, A / (?a + ?b). And even if you are faced with the task of "getting rid of the irrationality in the denominator." Then you can use the next method. Multiply the numerator and denominator of the fraction by the expression? a -? b. Thus, in the denominator, the formula for abbreviated multiplication will be obtained: (?a + ?b) * (?a - ?b) \u003d a - b. By analogy, if the difference of the roots is given in the denominator: ?a - ?b, then the numerator and denominator of the fraction must be multiplied by the expression? a + ?b. For example, let's say 4 / (?3 + ?5) = 4 * (?3 - ?5) / ((?3 + ?5) * (?3 - ?5)) = 4 * (?3 - ?5) / (-2) = 2 * (?5 - ?3).

4. Consider a more difficult example of getting rid of irrationality in the denominator. Let the fraction 12 / (?2 +?3 +?5) be given. You need to multiply the numerator and denominator of the fraction by the expression? 2 + ? 3 - ? 5:12 / (? 2 + ? 3 + ? 5) = 12 * (? + ?5) * (?2 + ?3 - ?5)) = 12 * (?2 + ?3 - ?5) / (2 * ?6) = ?6 * (?2 + ?3 - ?5) \u003d 2 *? 3 + 3 *? 2 -? 30.

5. And finally, if you only need an approximate value, then you can calculate the square roots on the calculator. Calculate the values ​​separately for the whole number and write down with the required precision (say, two decimal places). And then perform the required arithmetic operations, as with ordinary numbers. Say, let's say you need to find out the approximate value of the expression? 7 +? 5 ? 2.65 + 2.24 = 4.89.

Related videos

Note!
In no case can square roots be added as primitive numbers, i.e. ?3 + ?2? ?5!!!

Useful advice
If you factor out the number in order to move the square from under the root sign, then do the reverse check - multiply all the resulting factors and get the original number.

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. The addition of roots is similar to the addition of the terms of an algebraic expression, that is, it requires the definition of similar roots.

Steps

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.

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.

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). A cube root of 3 has no such roots.

• 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.

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Content:

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. The addition of roots is similar to the addition of the terms of an algebraic expression, that is, it requires the definition of similar roots.

Steps

Part 1 Finding Roots

1. 1 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 the 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 like this: 3 √(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 factor (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.
2. 2 Remember which roots can be folded and which cannot. Just as you cannot add different terms of an expression, for example, 2a + 2b ≠ 4ab, you cannot add different roots.
   • You cannot add roots with different radical expressions, for example, √(2) + √(3) ≠ √(5). But you can add the numbers under the same root, like √(2 + 3) = √(5) (the square root of 2 is about 1.414, the square root of 3 is about 1.732, and the square root of 5 is about 2.236) .
   • You cannot add roots with the same root expressions, but different exponents, for example, √ (64) + 3 √ (64) (this sum is not equal to 5 √ (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 Simplifying and Adding Roots

1. 1 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) + 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) + 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) + 3 √(81)
2. 2 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 the number 50 into 2*25 and the number 32 into 2*16. From 25 and 16, you can extract the 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) + 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 √(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). A cube root of 3 has no such roots.
   • 10√(2) + 4√(2) = 14√(2).
   • 2√(3)+ 6√(3) = 8√(3).
   • Final simplified expression: 14√(2) + 8√(3) + 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.