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At the beginning of the lesson we will review the basic properties square roots, and then consider a few complex examples to simplify expressions containing square roots.
Subject:Function. Properties of square root
Lesson:Converting and simplifying more complex expressions with roots
1. Review of the properties of square roots
Let us briefly repeat the theory and recall the basic properties of square roots.
Properties of square roots:
1. therefore, ;
3. 
;
4. 
.
2. Examples for simplifying expressions with roots
Let's move on to examples of using these properties.
Example 1: Simplify an expression 
.
Solution. To simplify, the number 120 must be factorized into prime factors:
We will reveal the square of the sum using the appropriate formula:
Example 2: Simplify an expression 
.
Solution. Let us take into account that this expression does not make sense for all possible values of the variable, since this expression contains square roots and fractions, which leads to a “narrowing” of the range of permissible values. ODZ: 
().
Let us reduce the expression in brackets to common denominator and write the numerator of the last fraction as a difference of squares:
Answer. at.
Example 3: Simplify an expression 
.
Solution. It can be seen that the second numerator bracket has an inconvenient appearance and needs to be simplified; let’s try to factor it using the grouping method.
To be able to calculate the common factor, we simplified the roots by factoring them. Let's substitute the resulting expression into the original fraction:
After reducing the fraction, we apply the difference of squares formula.
3. An example of getting rid of irrationality
Example 4. Free yourself from irrationality (roots) in the denominator: a) ; b) .
Solution. a) In order to get rid of irrationality in the denominator, the standard method of multiplying both the numerator and denominator of a fraction by the conjugate factor to the denominator is used (the same expression, but with the opposite sign). This is done to complement the denominator of the fraction to the difference of squares, which allows you to get rid of the roots in the denominator. Let's do this in our case:
b) perform similar actions:
4. Example for proof and identification of a complete square in a complex radical
Example 5. Prove equality 
.
Proof. Let's use the definition of a square root, from which it follows that the square of the right-hand expression must be equal to the radical expression:
. Let's open the brackets using the formula for the square of the sum:
, we got the correct equality.
Proven.
Example 6. Simplify the expression.
Solution. This expression is usually called a complex radical (root under root). IN in this example you need to guess to isolate a complete square from the radical expression. To do this, note that of the two terms, it is a candidate for the role of double the product in the formula for the squared difference (difference, since there is a minus). Let us write it in the form of the following product: , then the role of one of the terms full square claims , and for the role of the second - 1.
Let's substitute this expression under the root.
In 8th grade, schoolchildren in mathematics lessons are introduced to the concept of “radical” or, simply put, “root”. It was then that they first encountered the problem of simplifying complex radicals. Complex radicals are expressions in which one root is under another. Therefore, they are sometimes called nested radicals. In this article, the mathematics and physics tutor talks in detail about how to simplify a complex radical.
Methods for simplifying complex radicals
To simplify a complex radical means to get rid of the outer root. It is best to start studying this topic by simplifying double radicals. After all, if we learn to simplify double radicals, then we will also be able to simplify more complex ones.
How do we get rid of the outer root? It is clear that for this you need to transform the radical expression, presenting it in the form of a complete square. To do this, we will use the well-known formula “Square of the difference”:
Here, as you can see, the negative term has a factor on the right. Therefore, let's get this factor under the root. To do this, we present it as a product of:
Then and. It remains only to pay attention to the fact that 
. Now we can see that under the root we have a squared difference:
Now let's remember that. Exactly the module. This is very important here because the square root is a positive number. Then we get:
Well, since title="Rendered by QuickLaTeX.com" height="21" width="61" style="vertical-align: -3px;">, модуль раскрывается со знаком минус. В результате в ответе получаем:!}
This is how we managed to simplify this radical. But there are also more complex cases when it is not immediately possible to guess how to represent a radical expression in the form of a complete square. For example, in the following example.
In order not to rack your brains for a long time, you can use the following method.
Let me remind you that our goal is to represent the expression under the root as a perfect square. Specifically in this example, in the form of a square of the sum:
Well, the square of the sum is revealed according to the well-known formula, which we already wrote today:
So, the idea, in fact, is to take the irrational part of the radical expression for, and the rational part for. Then we get the following system of equations:
It is clear that . Otherwise, the second equation of the system is not satisfied. Then we express the coefficient from the second equation:
The denominator of this fraction is not equal to zero, which means its numerator is equal to zero. We obtain a biquadratic equation, which can be solved in the standard way (for more details, see the attached video). Solving it, we get as many as 4 roots. You can take any one. I like it better. Then . So, we finally get:
Here is a way to simplify a complex radical. There is one more. For those who like to remember complex formulas, which I am not. But for the sake of completeness, I’ll tell you about him too.
Formula of complex radicals
This is what the formula looks like:
Quite scary, isn't it? But don't be afraid, it can actually be used successfully in some cases. Let's look at an example:
We substitute the corresponding values into the formula:
This is the answer.
So, today in class I talked about how to simplify a complex radical. If you did not previously know the methods discussed today, then most likely you still have a lot to learn in order to feel confident on the Unified State Exam or the entrance exam in mathematics. But don't worry, I can teach you all this. All the necessary information about my classes is on. Good luck to you!
Material prepared by Sergey Valerievich