Transformation of expressions containing radicals is independent. Irrational expressions (expressions with roots) and their transformation. Problems to solve independently

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Practical lesson

Topic: Converting numeric and alphabetic expressions containing radicals.

Goals :

Educational: continue to develop students’ skills in applying the properties of powers and roots when transforming expressions.

Educational: fostering independence and a creative approach to problem solving.

Developmental: development of logical thinking, comparative analysis skills.

Equipment: board, computer, projector, screen, notes on the board, posters with formulas on the topic: “Degrees” and “Roots”, individual task cards.

Using elements of pedagogical technologies:

1. cooperation;

2. health-saving (alternating types of activities);

3. information and communication;

4. developing;

5. person-oriented.

Performance:

formation of competencies: value-semantic, educational-cognitive, communicative, personal self-improvement.

Lesson plan.

1) Preparatory stage.

1) Checking the assimilation of the material covered frontally (or individually) on the following questions (questions are projected on the screen, to which students answer orally).

1. What does it mean to raise a number to the power n?

2. How to multiply two powers with the same bases?

3. How to divide two powers with the same bases?

4. How to raise a degree to a power?

5. How to extract the root of a degree?

6. What is the zero power of a number?

7. How to find a degree with a negative exponent?

9. How to find a root with a fractional exponent?

10. Formulate the main property of the root.

11. How to extract the root from a product?

12. How to extract the root of a fraction?

13. How to extract the root of a degree?

14. How is the multiplication of roots of the same degree carried out?

15. How is the multiplication of roots of different degrees carried out?

16. How are roots of the same degree divided?

17. How is a root raised to a power?

2) Repeat:

properties of roots	properties of degrees

2) Theoretical stage.

Application of knowledge when solving typical tasks.

Exercise 1. Reduce the roots to a general indicator:

Task 2.

Task 3. Extract the root:

Task 4. Follow these steps:

Exercise5 . Calculate:

3) Practical stage.

Independent application of skills and knowledge.

Carry out independent work in 15 options.

1. Reduce the roots to a general indicator:

2. Reduce the indicators of roots and radical expressions:

3. Extract the root:

4. Follow these steps:

5. Calculate:

Bibliography.

1. Alimov Sh.A. and etc. Mathematics: algebra and principles of mathematical analysis, geometry. Algebra and beginnings of mathematical analysis (basic and advanced levels). 10-11 grades. - M., 2014.

2. Bogomolov N.V. Mathematics: textbook for applied bachelor's degree / N.V. Bogomolov, P.I. Samoilenko. – 5th ed., revised. and additional – M.: Yurayt Publishing House, 2014.

Lesson and presentation on the topic: "Converting expressions containing a radical"

Additional materials
Dear users, do not forget to leave your comments, reviews, wishes! All materials have been checked by an anti-virus program.

Teaching aids and simulators in the Integral online store for grade 11
Interactive manual for grades 9–11 "Trigonometry"
Interactive manual for grades 10–11 "Logarithms"

Guys, in the last lesson we studied the properties of the nth root. Today we will look at how to apply them in solving various problems that may arise in practice.

Let's make a little reminder of the properties of our roots:
1. $((\sqrt[n](a)))^n=a$; $\sqrt[n](a^n)=a$.
2. $\sqrt[n](a*b)=\sqrt[n](a)*\sqrt[n](b)$.
3. $\sqrt[n](\frac(a)(b))=\frac(\sqrt[n](a))(\sqrt[n](b))$, $b≠0$.
4. $((\sqrt[n](a)))^k=\sqrt[n](a^k)$.
5. $\sqrt[n](\sqrt[k](a))=\sqrt(a)$.
6. $\sqrt(a^(k*p))=\sqrt[n](a^(k))$.

Using our formulas, we can transform expressions containing radicals (root extraction operation); such expressions are called irrational.

Example.
Simplify the expression:
a) $\sqrt(48a^7)$.
b) $((\sqrt(a^3)))^2$.
Solution.
a) Let us reduce the radical expression to the form: $16*a^4*3a^3$.
Then, using formula 2 from our memo, the original expression will take the form:
$\sqrt(48a^7)=\sqrt(16*a^4*3a^3)=\sqrt(16)*\sqrt(a^4)*\sqrt(3a^3)=2a*\sqrt( 3a^3)$.
The expression we obtained is considered simpler, since the root sign has a simpler expression.
A transformation of this type is called taking the multiplier out of the radical sign.

B) Let's use formula 4: $((\sqrt(a^3)))^2=\sqrt(((a^3))^2)=\sqrt(a^6)$.
Let's transform the resulting expression using the same method as in the first example. $\sqrt(a^6)=\sqrt(a^5*a)=\sqrt(a^5)*\sqrt(a)=a*\sqrt(a)$.
When placing a multiplier outside the sign of the radical, special attention should be paid to the sign of the factor being removed. In the case of even powers, it can be either positive or negative.

Let's look at an example: $\sqrt(x^6*y)$.
We don’t know anything about the sign of the number x; transforming our expression we get: $x*\sqrt(y)$.
In fact, this entry is incorrect. We repeat: we know nothing about the sign of the number x. How to be in this case?
To be sure that the answer is correct, it is better to present it in the form: $|x|*\sqrt(y)$.
The generalized formula for roots with an even exponent will look like this: $\sqrt(a^(2n))=|a|$.

Guys, we looked at the operation of moving the multiplier beyond the radical sign. There is also a reverse operation - introducing a multiplier under the sign of the radical.

Example.
Compare the numbers $4\sqrt(2)$ and $2\sqrt(4)$.
Solution.
We know: $4=\sqrt(64)$ and $2=\sqrt(8)$.
Let's transform the original expression:
$4\sqrt(2)=\sqrt(64)*\sqrt(2)=\sqrt(128)$.
$2\sqrt(4)=\sqrt(8)*\sqrt(4)=\sqrt(32)$.
The roots of both expressions are the same. The number whose radical expression is larger is larger. In our case: $\sqrt(128)>\sqrt(32)$.

Example.
Simplify the expression: $\sqrt(x^3*\sqrt(x))$.
Solution.
Let's enter an expression containing the third degree under the sign of the root:
$x^3*\sqrt(x)=\sqrtx^(12)*\sqrt(x)=\sqrt(x^(13))$.
Let's use formula 5. The original expression can be represented as: $\sqrt(\sqrt(x^(13)))=\sqrt(x^(13))$.

Example.
Follow these steps:
a) $(\sqrt(a)-\sqrt(b))(\sqrt(a)+\sqrt(b))$.
b) $(\sqrt(a)-\sqrt(b))(\sqrt(a^2)+\sqrt(ab)+\sqrt(b^2))$.
Solution:
a) Let's use the difference of squares formula:
$(\sqrt(a)-\sqrt(b))(\sqrt(a)+\sqrt(b))=(\sqrt(a^2)+\sqrt(b^2))$.
Now let's simplify the expression we received, use formula 6 of our memo:
$(\sqrt(a^2)-\sqrt(b^2))=(\sqrt(a)-\sqrt(b))$ (the root exponent and the degree of the radical expression were divided by 2.
Answer: $(\sqrt(a^2)-\sqrt(b^2))(\sqrt(a^2)+\sqrt(b^2))=(\sqrt(a)-\sqrt(b) )$.

B) Let's look carefully at our expression. It's similar to the difference of cubes formula, so let's apply it:
$(\sqrt(a)-\sqrt(b))(\sqrt(a^2)+\sqrt(ab)+\sqrt(b^2))=((\sqrt(a)))^3- ((\sqrt(b)))^3=a-b$.

Example.
Follow these steps:
a) $\sqrt(a^5)*\sqrt(a^3)$.
b) $\sqrt(3-\sqrt(3))*\sqrt(12+6\sqrt(3))$.
Solution.
You can only multiply roots of the same degree. Let's reduce our expressions to the same root exponent.
$\sqrt(a^5)=\sqrt(a^(10))$ (multiplied by 2).
$\sqrt(a^3)=\sqrt(a^(9))$ (multiplied by 3).
$\sqrt(a^5)*\sqrt(a^3)=\sqrt(a^(10))*\sqrt(a^9)=\sqrt(a^(19))$.
Let's simplify the resulting expression:
$\sqrt(a^(19))=\sqrt(a^(12)*a^7)=|a|*\sqrt(a^7)$.
Let us pay attention to the fact that the exponent of the root of our expressions is even. This means that the radical expression contains only positive numbers, that is, $a≥0$, but then $|a|=a$.
Answer: $\sqrt(a^5)*\sqrt(a^3)=a*\sqrt(a^7)$.

B) This example can be solved in two ways. Let's look at each method:
1 way. Let's bring the first factor to the 4th power:
$\sqrt(3-\sqrt(3))=\sqrt(((3-\sqrt(3)))^2)=\sqrt(9-6\sqrt(3)+3)=\sqrt(12 -6\sqrt(3))$.
Let's multiply the radicals:
$\sqrt(12-6\sqrt(3))*\sqrt(12+6\sqrt(3))=\sqrt(((12-6\sqrt(3)))*(12+6\sqrt( 3)))=\sqrt(144-36*3)=\sqrt(144-108)=\sqrt(36)=\sqrt(6^2)=\sqrt(6)$.

Method 2. Let's look at the radical expression in the second factor:
$12+6\sqrt(3)=9+6\sqrt(3)+3=3^2+2*3*\sqrt(3)+((\sqrt(3)))^2=((3+ \sqrt(3)))^2$.
We can convert the multiplier as a whole:
$\sqrt(12+6\sqrt(3))=\sqrt(((3+\sqrt3))^2)=\sqrt(3+\sqrt(3))$ (divided by 2 exponents).
Let's transform the entire expression:
$\sqrt(3-\sqrt(3))*\sqrt(12+\sqrt(3))=\sqrt(3-\sqrt(3))*\sqrt(3+\sqrt(3))=\ sqrt((3-\sqrt(3))*(3+\sqrt(3)))=\sqrt(9-3)=\sqrt(6)$.

Example.
Factor the expression: $\sqrt(x^8)-2\sqrt(x^4y^2)+\sqrt(y^4)$.
Solution.
Let's rewrite the original expression as:
$\sqrt(x^8)-2\sqrt(x^4y^2)+\sqrt(y^4)=((\sqrt(x^4)))^2-2*\sqrt(x^4 )*\sqrt(y^2)+((\sqrt(y^2)))^2$ is the so-called “squared difference”.
$\sqrt(x^8)-2\sqrt(x^4y^2)+\sqrt(y^4)=((\sqrt(x^4)-\sqrt(y^2)))^2= ((x\sqrt(x)-\sqrt(y^2)))^2$.

Example.
Reduce the fraction: $\frac(\sqrt(x)-\sqrt(y))(\sqrt(x)-2\sqrt(xy)+\sqrt(y))$.
Solution.
1 way.
Let's look at the numerator and denominator separately:
$\sqrt(x)-\sqrt(y)=\sqrt(x^2)-\sqrt(y^2)=(\sqrt(x)-\sqrt(y))(\sqrt(x)+\ sqrt(y))$.
$\sqrt(x)-2\sqrt(xy)+\sqrt(y)=\sqrt(x^2)-2\sqrt(xy)+\sqrt(y^2)=((\sqrt(x) -\sqrt(y)))^2$.
Let's shorten the resulting expression:
$\frac(\sqrt(x)-\sqrt(y))(\sqrt(x)-2\sqrt(xy)+\sqrt(y))$=$\frac((\sqrt(x)-\ sqrt(y))(\sqrt(x)+\sqrt(y)))(((\sqrt(x)-\sqrt(y)))^2)$=$\frac(\sqrt(x)+ \sqrt(y))(\sqrt(x)-\sqrt(y))$.

Method 2.
Let us introduce a change of variables.
Let $a=\sqrt(x)$, $b=\sqrt(y)$. Then $\sqrt(x)=a^2$ and $\sqrt(y)=b^2$.
$\frac(\sqrt(x)-\sqrt(y))(\sqrt(x)-2\sqrt(xy)+\sqrt(y))=\frac(a^2-b^2)(( a^2-2ab+b)^2)=\frac((a-b)(a+b))(((a-b)^2))=\frac((a+b))((a-b))=\ frac(\sqrt(x)+\sqrt(y))(\sqrt(x)-\sqrt(y))$.
Changing variables often simplifies the solution. Working with rational expressions is much easier and more familiar than with irrational ones.

Problems to solve independently

1. Simplify the expression:
a) $\sqrt(162a^5)$.
b) $((\sqrt(a^5)))^3$.
2. Compare the numbers: $3\sqrt(4)$ and $2\sqrt(5)$.
3. Simplify the expression: $\sqrt((x^2)*\sqrt(x^2))$.
4. Follow these steps:
a) $\sqrt(a^7)*\sqrt(a^4)$.
b) $\sqrt(4-\sqrt(3))*\sqrt(19+8\sqrt(3))$.
5. Factor the expression: $\sqrt(x^6)-6\sqrt(x^3y^5)+9\sqrt(y^(10))$.
6. Reduce the fraction: $\frac(\sqrt(x)-\sqrt(y))(\sqrt(x)+2\sqrt(xy)+\sqrt(y))$.

Lesson topic:

Converting expressions containing radicals.

The purpose of the lesson:

Educational:

Formation of ability to solvetasks to transform expressions containing radicals;

consolidate the concepts of root propertiesn-Ouch;

contribute to the improvement of skills and abilities to work inMicrosoftOfficeExcelwhen processing information in production.

Developmental:

development of thinking skills: structuring objects (identifying the component parts of an object and arranging them in a hierarchical form).

develop creative (productive) thinking (in the process of composing a puzzle),

Educational:

nurturing a general and information culture, hard work, perseverance, patience, careful attitude to computer technology, instilling in students the skills of independence in work.

Lesson type: systematization of knowledge

Lesson type: problem

Methodical techniques: visual - illustrative: rebus, computer testing, practical: selective solution of examples, production tasks

Equipment and visual aids for teaching: computer class with Windows XP OS and Microsoft Office 2003 software package, multimedia projector, presentation, computer test, handouts (rebus).

Interdisciplinary connections: mathematics - computer science - industrial training.

During the classes:

I .Organizing time: Preparing students for the lesson

(checking absences from class, availability of notebooks), communicating the topic and goals

lesson. Slide1,2

Motivation.

II .Updating basic knowledge:

2.1 Frontal survey:

2.2.1 What is a radical? Slide 5.

2.2.3 List:

a) properties of the nth root. Slide 6.

b) the root of a fraction. Slide7.

c) Extracting the root from the root. Slide 8.

d) the main property of the root. Slide 9.

III . Practical work.

Solve the examples. Based on the answer in the example, select the corresponding letter in the rebus, write the answer in the table. The resulting term “----” is an organized sequence of actions.

V . Summing up the lesson:

Today in class we confirmed the words of the Russian scientist M.V. Lomonosov

“ Let someone try to eliminate degrees from mathematics, and he will see that without them you won’t get far.”(M.V. Lomonosov) . Without radicals, it is not possible to calculate the energy costs of an enterprise. And by studying at this lyceum in the profession of “Computer Operator” and receiving information about working with computer equipment in industrial training classes, you can process any information using information technology. Therefore, the words of Nathan Rothschild “Who owns the information, owns the world” are very relevant when working in your profession in any enterprise or factory.

Grading for the lesson.

V .Homework:

We call algebraic expressions that use not only four rational actions, but also radical signs (from literal expressions) irrational algebraic expressions.

These are, for example, the expressions

When determining o. d.z. For irrational algebraic expressions, it should be taken into account that expressions under the sign of a radical of even degree must not be negative. When finding the numerical values ​​of an expression for given literal values ​​of the parameters, roots of even degree are understood in the arithmetic sense.

Example 1. Find o. d.z. expressions

and its value at .

Solution. O.d.z. determined from the conditions. We find that Fr. d.z. is determined by inequalities. When calculating the value at a given point we get

When transforming irrational algebraic expressions, all the rules for operations with roots are used (Chapter I, § 2). Let us first consider possible simplifications of expressions such as “the root of a monomial” or “the root of the quotient of two monomials.” We will say that a root is reduced to its simplest form if: 1) it does not contain irrationality in the denominator, 2) it is impossible to reduce its exponent with the exponent of the radical expression, and, finally, 3) all possible factors are removed from under the root. Any given root can be reduced to its simplest form, that is, replaced by an identically equal one, but one that meets all three listed conditions.

Example 2. Reduce the following roots to their simplest form:

Solution, a) Reduce by 3 the exponent of the root and the exponent of each of the factors of the radical expression

We take out the factors a and ; from under the root sign.

Roots whose simplest forms differ, perhaps only in coefficients (numerical or alphabetic), are usually called similar. For example, the roots and are similar, since and the roots are not similar, since

When adding and subtracting similar roots, they are all reduced to their simplest form, and then the root is taken out of brackets.

Example 3. Perform the following actions:

Solution. Let's reduce each of the roots to its simplest form:

Now we find (all roots turned out to be similar)

When removing factors from under the sign of a root of an even degree, it is necessary to remember that the root is understood in an arithmetic sense. So, if the signs a, b are not indicated, then you should write not. Here about. d.z. consists not only of values ​​, but also of values ​​a

Example 4: Simplify an expression

The following cases are possible:

If you do not assume in advance that , then solving the example will become even more complicated, since you will have to write the answer in a general form:

and then consider four possible cases: . We leave this analysis to the reader to complete.

In the example we just solved, the radical expressions were represented as the exact squares of some binomials in an obvious way. In some cases, this representation of the radical expression is not done in such an obvious way. So, sometimes you can simplify radicals of the form