Presentation for the lesson "degree with rational exponent". Presentation “Degree with a rational exponent Presentation for the lesson exponent with a rational exponent

To use presentation previews, create a Google account and log in to it: https://accounts.google.com

Slide captions:

Degree with a rational exponent Definitions and properties of a degree with a rational exponent Elena Olegovna Reva. MBOU "Gymnasium No. 16" Mytishchi

Continue the formula:

A little history: 1. Find the meaning of the expression: What Latin letter did European mathematicians, starting from the 13th century, use to denote the root? and answer the following question: N, then N x K, then K x R, then R x 5.8 2 -5.8

Medieval mathematicians, such as the Italian scientist Gerolamo Cardano, denoted the square root with the symbol R or the stylized combination R x (from the Latin Radix - root). The figure shows how Cardano wrote down the equality in 1585: A little history: D. Cardano 1501-1576

A little history: 2. Simplify: Which mathematician introduced a root notation in 1626 that resembles modern notation? and find out the answer to the following question: Christophe Rudolf A l'bert Girard Simon Stevin

In 1626, a French mathematician living in the Netherlands, Albert Girard, introduced the use of the root symbol of an arbitrary power (before him, the radical symbol was used only for the square root). This designation began to replace the R sign. The plus-minus sign. A little history: A. Girard 1595–1632

A little history: 3. Simplify the expression After solving the problem, you will find out the answer to the following question: and find its value at x = 0.20 14 . Who was the first to use a line over a radical expression? René Descartes François Viet Thomas Herriot 14 0.4028 14.4028

A little history: At first there was no line above the radical expression; it was later introduced by Rene Descartes instead of parentheses. Only in 1637 R. Descartes connected the root sign with a horizontal line. The modern root sign finally came into general use only at the beginning of the 18th century. R. Descartes 1596 - 1650

Degree with a rational exponent Def.: Properties of degrees: Studying the theory:

No. 1. P about h i t a e m: a) b) No. 1. P o c h i t a e m:

c) d) No. 1. P o c h i t a e m: No. 1. P o c h i t a e m:

a) because x > 0 b) No. 2. We solve the equation:

Working independently #1. Simplify the expression: No. 2. Solve the equation: Recommendations: see textbook p.54 Example 2. Recommendations: see textbook p.55 Example 4.

We apply theory No. 3. Simplify the expression: Answer: Answer: No. 4. For what x is the equality true: (-  ;0 ] Recommendations: determine the sign of the left side...

Nowadays, quite often during lessons in schools, presentations, videos and other electronic resources are shown in parallel, with the help of which the learning process can be made more effective. Today there is a huge database of similar materials that can be downloaded on the Internet.

The presentation on "Exponent with Rational Exponent" is an excellent example of an e-learning resource. With its help you can create a good structured lesson summary on this topic. This will help the novice teacher not to get confused in the lesson and convey the material to each student.

By the 9th grade, students had already encountered the concept of degrees. The indicator of a power expression can be not only a natural or an integer expression. It can be a rational number or a rational expression. This is a big topic that deserves a lot of attention.

The presentation “Exponent with a rational exponent” contains 12 slides.

After the greeting, the first example of a power is shown, the exponent of which is the rational expression 1/n. The base of the degree is also a literal value, positive. At the same time, it is noted that the denominator of the indicator is a natural value. Such an entry can be replaced using the root sign. This is clearly demonstrated on this page. A teacher or tutor can comment and add examples with numerical values ​​so that students can remember it easier.

The next slide contains a question: how can you write a similar fraction through the root, which in the exponent contains the fractional expression m/n. The answer to this question lies on the next slide. This entry can be represented as a radical expression, where the numerator of the power expression is the degree of the radical expression, that is, a, and the denominator is the exponent of the radical expression.

The fifth slide is dedicated to demonstrating examples. Three cases are given where you can see degrees with rational exponents. Moreover, it is worth noting that they differ in the form of recording and signs. Both decimal fractions and ordinary fractions are written as indicators.

The next slide explains to the student how to find a power whose base is equal to zero. Whatever the indicator, the answer will be zero. This needs to be remembered. Below is the formula with the letter designation of the indicator.

The following pages are devoted to a discussion of the properties of degrees. They are similar for both integer and rational indicators.

First of all, three formulas are given. The first states that to multiply powers with the same bases, you must add the powers. The second demonstrates the division of similar degrees. And the third formula shows how you can raise a certain degree to a power. As you can see, to do this you need to multiply the indicators by each other.

The next slide provides formulas for exponentiation of a product and a quotient. This will often be encountered in the future when solving various equations and systems, or to simplify huge expressions, etc.

In the first formula, you can see that in order to raise the product of some values ​​of a and b, it is necessary to raise each value separately to the same power. The reverse expression is also true. This can be verified using a numerical example.

The last slides are dedicated to examples. To solve them, you need to have a good understanding of the essence of powers with a rational exponent and a good knowledge of their properties.

The first example contains a variable x, the value of which is specified in the condition. Before substituting it, it is necessary to simplify the expressions as much as possible. Once this procedure is completed, you can substitute the existing value in the condition for the unknown.

Example two is a fraction where both the numerator and denominator contain powers with rational exponents. These examples can be provided to 9th grade students for solution during independent work or tests. If students find it difficult to solve, you need to give them a hint that both the numerator and denominator need to be factored.

This presentation is very coherent and clear. It does not contain unnecessary illustrations and extended theory. Thanks to it, you can explain to a 9th grader in a very clear way about powers with rational exponents.

The educational material will be useful for both beginning tutors and experienced ones.

Degree With rational indicator

Completed by: Mathematics teacher of OGBPOU "RPTK"

Lukyanova A.P.

1 . The nth root and its properties

1.1. The definition of the nth root is given. Replace the numbers with these words to get the correct definition:

The nth root of a number a is a ① whose ②th power is equal to ③ .

Answers:

① - number,

② - nth degree

③ - A

1.2. Find the values:

Does not exist

1.3. Select correct equalities and correct errors in incorrect equalities

A);

b) ;

c) ;

d)

e)

Answer:

correct a,d;

incorrect b, c, e

Right:

Irrational equations

1.4. Find the roots of the equation:

(Answer: 18)

1.5. Check which numbers are 0; -2; 4 are the roots of the following equation:

(Answer: 4)

Comparison of nth roots

1.6. Arrange the numbers in ascending order:

Answer: ; ;

0 with a rational exponent r=m/n, where m is an integer, n is a natural number (n1)? A); b); c); d) 2.2. Add the properties: For any rational numbers r and s and any positive a and b, the following equalities hold: 1) 2) 3) 4) 5) 2.3. Compare the numbers: a) and; b) and; c)and; d) and " width="640"

Power with rational exponent

2.1. What is the definition of a power of a number? A 0 with a rational exponent r=m/n, where m is an integer, n is a natural number (n1)?

A); b); c); d)

2.2. Add the properties: For any rational numbers r and s and any positive a and b the equalities are valid:

1)

2)

3)

4)

5)

2.3. Compare the numbers: a) and; b) and ; c)and; d) and

Exercises

3.1 Find the meanings of the expressions:

A) ;

b) ;

V)

Exercises

3.2 Factor into:

A) ; b)

3.3 Simplify the expressions:

• b ) +

Homework

Find the meanings of the expressions:

a)2; b)

Factor it out:

A) ; b)

Simplify the expression:

Presentation for the lesson “Exponent with rational exponent”

Goals:

• educational: independent study of new material;
• educational: fostering interest in the subject, mathematical culture;
• developmental: development of independence, ability to acquire knowledge.

View document contents
“Presentation for the lesson “Exponent with a rational exponent””

Lesson topic: “Exponent with rational exponent”

Lesson objectives:

1) educational: independent study of new material;

2) educational: nurturing interest in the subject , mathematical culture;

3) developing: development of independence, ability to acquire knowledge.

Working with signal cards

Working with signal cards (green, red). I read out the statements. If the statement is true, then they show a green card, if it is false, a red card.

Independent study of new material Part 1 1. Consider the problem. 2. Draw a conclusion. 3. Consider 2-3 examples from the textbook. Part 2. 1. Consider the properties of the degree. 2. Consider examples in the textbook on the use of properties of degree. Part 3. Question-problem: why is a power with a rational exponent defined only for any positive base a?

The work is carried out according to the textbook using a plan. Students independently read theoretical material in the textbook and consider examples, make notes in notebooks, according to the plan. Then a survey is conducted on the material reviewed. When answering, you can use notes in notebooks and a textbook.

Reinforcing the material learned Task 1. Continue the phrase or fill in the blanks 1) A degree with a rational exponent is defined only for ... base a. 2) A degree with a rational exponent can be represented as…. Give examples. 3) The root can be represented as…. Give examples. 4) All properties of a degree with a natural exponent are true for a degree with ... exponent and ... base. 5) When multiplying powers with the same bases, the base is ..., and the exponents are .... 6) When dividing powers with the same bases, the base is ..., and the exponents are ....

Checking theoretical material reviewed by students independently.

Task 2. Solve examples.

The work is carried out orally along the chain.

Task 3. Calculate.

Performed at the board.

Task 4. Calculate.

Students comment on the solution from their seats.

Task 5. Independent work.

Students work independently (according to options) followed by checking on slide No. 10.

Answers to task 5. Option 1. Answer: 0.3. Option 2. Answer: 3. Option 3. Answer: 1.3. Option 4. Answer: 2.7.

Task 6. Calculate. Think about how you can solve these examples if the exponent is an irrational number.

Additional task. Work in pairs followed by mutual testing.

Summing up the lesson - What topic did you learn about in class? - What new did you learn? -What is not entirely clear in solving the examples? -Who needs consultation after classes?

The consultation is conducted by strong students and a teacher.

Homework Learn theoretical material.

Thanks everyone for the lesson

Goodbye