How to solve a degree with a real exponent. Degree with rational and real exponent

Degree with rational exponent

The set of rational numbers includes integers and fractional numbers.

Definition 1

Power of a number $a$ with integer exponent $n$ is the result of multiplying the number $a$ by itself $n$ times, and: $a^n=a \cdot a \cdot a \cdot \ldots \cdot a$, for $n>0$; $a^n=\frac(1)(a \cdot a \cdot a \cdot \ldots \cdot a)$, for $n

Definition 2

Power of a number $a$ with fractional exponent $\frac(m)(n)$ is called the $n$-th root of $a$ to the power of $m$: $a^\frac(m)(n)=\sqrt[n](a^m)$, where $a>0$, $ n$ is a natural number, $m$ is an integer.

Definition 3

Power of zero with fractional exponent $\frac(m)(n)$ is defined as follows: $0^\frac(m)(n)=\sqrt[n](0^m)=0$, where $m$ is an integer, $m>0$, $n$ is a natural number.

There is another approach to determining the degree of a number with a fractional exponent, which shows the possibility of the existence of a degree of a negative number or a negative fractional exponent.

For example, the expressions $\sqrt((-3)^6)$, $\sqrt((-3)^3)$ or $\sqrt((-7)^(-10))$ make sense, thus and the expressions $(-3)^\frac(6)(7)$, $(-3)^\frac(3)(7)$ and $(-7)^\frac(-10)(6)$ should make sense, while, according to the definition of a degree with an exponent in the form of a fraction with a negative base, they do not exist.

Let's give another definition:

Power of $a$ with fractional exponent $\frac(m)(n)$ is called $\sqrt[n](a^m)$ in the following cases:

For any real number $a$, integer $m>0$ and odd positive integer $n$.

For example, $13.4^\frac(7)(3)=\sqrt(13.4^7)$, $(-11)^\frac(8)(5)=\sqrt((-11)^8 )$.

For any non-zero real number $a$, integer negative $m$ and odd $n$.

For example, $13.4^\frac(-7)(3)=\sqrt(13.4^(-7))$, $(-11)^\frac(-8)(5)=\sqrt(( -11)^(-8))$.

For any non-negative number $a$, positive integer $m$ and even $n$.

For example, $13.4^\frac(7)(4)=\sqrt(13.4^7)$, $11^\frac(3)(16)=\sqrt(11^3)$.

For any positive $a$, integer negative $m$ and even $n$.

For example, $13.4^\frac(-7)(4)=\sqrt(13.4^(-7))$, $11^\frac(-3)(8)=\sqrt(11^(-3 ))$.

Under other conditions, the degree with a fractional indicator cannot be determined.

For example, $(-13,4)^\frac(10)(3)=\sqrt((-13,4)^(10))$, $(-11)^\frac(5)(4)= \sqrt((-11)^5)$.

In addition, when applying this definition, it is important that the fractional exponent $\frac(m)(n)$ be an irreducible fraction.

The seriousness of this remark is that the degree of a negative number with a fractional reduced exponent, for example, $\frac(10)(14)$ will be a positive number, and the degree of the same number with an already reduced exponent $\frac(5)(7)$ will be a negative number.

For example, $(-1)^\frac(10)(14)=\sqrt((-1)^(10))=\sqrt(1^(10))=1$ and $(-1)^ \frac(5)(7)=\sqrt((-1)^5)=-1$.

Thus, when the fraction reduction $\frac(10)(14)=\frac(5)(7)$ is performed, the equality $(-1)^\frac(10)(14)=(-1)^\ frac(5)(7)$.

Remark 1

It should be noted that a more convenient and simple first definition of the degree with an exponent in the form of a fraction is more often used.

In the case of writing a fractional exponent as a mixed fraction or decimal, it is necessary to convert the exponent to the form of an ordinary fraction.

For example, $(2 \frac(3)(7))^(1 \frac(2)(7))=(2 \frac(3)(7))^\frac(9)(7)=\sqrt ((2 \frac(3)(7))^9)$, $7^(3,6)=7^\frac(36)(10)=\sqrt(7^(36))$.

Degree with irrational and real exponent

To valid numbers include rational and irrational numbers.

Let us analyze the concept of degree with an irrational exponent, since degree with a rational exponent we have considered.

Consider a sequence of approximations to the number $\alpha$, which are rational numbers. Those. we have a sequence of rational numbers $\alpha_1$, $\alpha_2$, $\alpha_3$, $\ldots$, which determine the number $\alpha$ with any degree of accuracy. If we calculate the powers with these exponents $a^(\alpha_1)$, $a^(\alpha_2)$, $a^(\alpha_3)$, $\ldots$, then it turns out that these numbers are approximations to some number $ b$.

Definition 4

Power $a>0$ with irrational exponent $\alpha$ is an expression $a^\alpha$ that has a value equal to the limit of the sequence $a^(\alpha_1)$, $a^(\alpha_2)$, $a^(\alpha_3)$, $\ldots$, where $ \alpha_1$, $\alpha_2$, $\alpha_3$, … are successive decimal approximations of the irrational number $\alpha$.

In this article, we will understand what is degree of. Here we will give definitions of the degree of a number, while considering in detail all possible exponents of the degree, starting with a natural exponent, ending with an irrational one. In the material you will find a lot of examples of degrees covering all the subtleties that arise.

Page navigation.

Degree with natural exponent, square of a number, cube of a number

Let's start with . Looking ahead, let's say that the definition of the degree of a with natural exponent n is given for a , which we will call base of degree, and n , which we will call exponent. Also note that the degree with a natural indicator is determined through the product, so to understand the material below, you need to have an idea about the multiplication of numbers.

Definition.

Power of number a with natural exponent n is an expression of the form a n , whose value is equal to the product of n factors, each of which is equal to a , that is, .
In particular, the degree of a number a with exponent 1 is the number a itself, that is, a 1 =a.

Immediately it is worth mentioning the rules for reading degrees. The universal way to read the entry a n is: "a to the power of n". In some cases, such options are also acceptable: "a to the nth power" and "nth power of the number a". For example, let's take the power of 8 12, this is "eight to the power of twelve", or "eight to the twelfth power", or "twelfth power of eight".

The second power of a number, as well as the third power of a number, have their own names. The second power of a number is called the square of a number, for example, 7 2 is read as "seven squared" or "square of the number seven". The third power of a number is called cube number, for example, 5 3 can be read as "five cubed" or say "cube of the number 5".

It's time to bring examples of degrees with physical indicators. Let's start with the power of 5 7 , where 5 is the base of the power and 7 is the exponent. Let's give another example: 4.32 is the base, and the natural number 9 is the exponent (4.32) 9 .

Please note that in the last example, the base of the degree 4.32 is written in brackets: to avoid discrepancies, we will take in brackets all the bases of the degree that are different from natural numbers. As an example, we give the following degrees with natural indicators , their bases are not natural numbers, so they are written in parentheses. Well, for complete clarity at this point, we will show the difference contained in the records of the form (−2) 3 and −2 3 . The expression (−2) 3 is the power of −2 with natural exponent 3, and the expression −2 3 (it can be written as −(2 3) ) corresponds to the number, the value of the power 2 3 .

Note that there is a notation for the degree of a with an exponent n of the form a^n . Moreover, if n is a multivalued natural number, then the exponent is taken in brackets. For example, 4^9 is another notation for the power of 4 9 . And here are more examples of writing degrees using the “^” symbol: 14^(21) , (−2,1)^(155) . In what follows, we will mainly use the notation of the degree of the form a n .

One of the problems, the reverse of exponentiation with a natural exponent, is the problem of finding the base of the degree from a known value of the degree and a known exponent. This task leads to .

It is known that the set of rational numbers consists of integers and fractional numbers, and each fractional number can be represented as a positive or negative ordinary fraction. We defined the degree with an integer exponent in the previous paragraph, therefore, in order to complete the definition of the degree with a rational exponent, we need to give the meaning of the degree of the number a with a fractional exponent m / n, where m is an integer and n is a natural number. Let's do it.

Consider a degree with a fractional exponent of the form . In order for the property of degree in a degree to remain valid, the equality must hold . If we take into account the resulting equality and the way we defined , then it is logical to accept, provided that for given m, n and a, the expression makes sense.

It is easy to check that all properties of a degree with an integer exponent are valid for as (this is done in the section on the properties of a degree with a rational exponent).

The above reasoning allows us to make the following conclusion: if for given m, n and a the expression makes sense, then the power of the number a with a fractional exponent m / n is the root of the nth degree of a to the power m.

This statement brings us close to the definition of a degree with a fractional exponent. It remains only to describe for which m, n and a the expression makes sense. Depending on the restrictions imposed on m , n and a, there are two main approaches.

The easiest way to constrain a is to assume a≥0 for positive m and a>0 for negative m (because m≤0 has no power of 0 m). Then we get the following definition of the degree with a fractional exponent.

Definition.

Power of a positive number a with fractional exponent m/n, where m is an integer, and n is a natural number, is called the root of the nth of the number a to the power of m, that is, .

The fractional degree of zero is also defined with the only caveat that the exponent must be positive.

Definition.

Power of zero with fractional positive exponent m/n, where m is a positive integer and n is a natural number, is defined as .
When the degree is not defined, that is, the degree of the number zero with a fractional negative exponent does not make sense.

It should be noted that with such a definition of the degree with a fractional exponent, there is one nuance: for some negative a and some m and n, the expression makes sense, and we discarded these cases by introducing the condition a≥0 . For example, it makes sense to write or , and the above definition forces us to say that degrees with a fractional exponent of the form are meaningless, since the base must not be negative.

Another approach to determining the degree with a fractional exponent m / n is to separately consider the even and odd exponents of the root. This approach requires an additional condition: the degree of the number a, whose exponent is , is considered the degree of the number a, the exponent of which is the corresponding irreducible fraction (the importance of this condition will be explained below). That is, if m/n is an irreducible fraction, then for any natural number k the degree is first replaced by .

For even n and positive m, the expression makes sense for any non-negative a (the root of an even degree from a negative number does not make sense), for negative m, the number a must still be non-zero (otherwise division by zero will occur). And for odd n and positive m, the number a can be anything (the root of an odd degree is defined for any real number), and for negative m, the number a must be different from zero (so that there is no division by zero).

The above reasoning leads us to such a definition of the degree with a fractional exponent.

Definition.

Let m/n be an irreducible fraction, m an integer, and n a natural number. For any reducible ordinary fraction, the degree is replaced by . The power of a with an irreducible fractional exponent m / n is for



Let us explain why a degree with a reducible fractional exponent is first replaced by a degree with an irreducible exponent. If we simply defined the degree as , and did not make a reservation about the irreducibility of the fraction m / n , then we would encounter situations similar to the following: since 6/10=3/5 , then the equality , but , a .

Lesson topic: A degree with a real exponent.

Tasks:

• Educational:
  • generalize the concept of degree;
  • to develop the ability to find the value of the degree with a real indicator;
  • consolidate the ability to use the properties of the degree when simplifying expressions;
  • develop the skill of using the properties of the degree in calculations.
• Educational:
  • intellectual, emotional, personal development of the student;
  • develop the ability to generalize, systematize on the basis of comparison, draw a conclusion;
  • activate independent activity;
  • develop curiosity.
• Educational:
  • education of communicative and informational culture of students;
  • aesthetic education is carried out through the formation of the ability to rationally, accurately draw up a task on a blackboard and in a notebook.

Students should know: definition and properties of a degree with a real exponent.

Students should be able to:

• determine whether an expression with a degree makes sense;
• use the properties of the degree in calculations and simplification of expressions;
• solve examples containing a degree;
• compare, find similarities and differences.

Lesson form: seminar - workshop, with elements of research. Computer support.

Form of organization of training: individual, group.

Lesson type: lesson of research and practical work.

DURING THE CLASSES

Organizing time

“One day the king decided to choose his first assistant from among his courtiers. He led everyone to a huge castle. "Whoever opens it first will be the first helper." No one even touched the castle. Only one vizier came up and pushed the lock, which opened. It was not locked.
Then the king said: “You will receive this position because you rely not only on what you see and hear, but rely on your own strength and are not afraid to make an attempt.”
And today we will try, try to come to the right decision.

1. What mathematical concept is associated with the words:

Base
Index (Degree)
What words can combine the words:
rational number
Integer
Natural number
irrational number (Real number)
Formulate the topic of the lesson. (Power with real exponent)

2. What is our strategic goal? (USE)
What kind objectives of our lesson?
- Generalize the concept of degree.

Tasks:

- repeat the properties of the degree
– consider the use of degree properties in calculations and simplifications of expressions
- development of computational skills.

3. So, a p, where p is a real number.
Give examples (choose from expressions 5 -2, 43,) degrees

- with a natural indicator
- with integer value
- with a rational indicator
- with an irrational indicator

4. At what values a the expression makes sense

an, where n (a is any)
am, where m (a 0) How to move from a degree with a negative exponent to a degree with a positive exponent?
, where (a0)

5. From these expressions, choose those that do not make sense:
(–3) 2 , , , 0 –3 , , (–3) –1 , .
6. Calculate. The answers in each column share one common property. Specify an extra answer (does not have this property)

2 = =
= 6 = (wrong others) = (cannot write dec. others)
= (fraction) ==

7. What actions (mathematical operations) can be performed with degrees?

Set match:

One student writes down formulas (properties) in general terms.

8. Supplement the degrees from item 3 so that the properties of the degree can be applied to the resulting example.

(One person works at the blackboard, the rest in notebooks. To check, exchange notebooks, and one more performs actions on the board)

9. On the board (student works):

Calculate :=

Independently (with check on sheets)

Which of the answers cannot be obtained in part "B" on the exam? If the answer turned out, then how to write such an answer in part "B"?

10. Independent completion of the task (with a check at the blackboard - several people)

Multiple Choice Task

1
2	:
3	0,3
4

11. Task with a short answer (solution at the blackboard):

+ + (60)5 2 – 3–4 27 =

On your own with verification on a hidden board:

– – 322– 4 + (30)4 4 =

12 . Reduce the fraction (on the board):

At this time, one person decides on the board on their own: = (class checks)

13. Independent decision (for verification)

To mark "3": Test with a choice of answers:

1. Specify an expression equal to the degree

1.

2.

3.

4.

2. Present the product as a degree: - Thank you for the lesson!

For any angle α such that α ≠ πk/2 (k belongs to the set Z), we have:

For any angle α, the equalities are true:

For any angle α such that α ≠ πk (k belongs to the set Z), we have:

Cast formulas

The table contains reduction formulas for trigonometric functions.

Function (angle in º)	90º - α	90º + α	180º - α	180º + α	270º - α	270º + α	360º - α	360º + α
sin	cosα	cosα	sinα	-sinα	-cos α	-cos α	-sinα	sinα
cos	sinα	-sinα	-cos α	-cos α	-sinα	sinα	cosα	cosα
tg	ctgα	-ctgα	-tgα	tgα	ctgα	-ctgα	-tgα	tgα
ctg	tgα	-tgα	-ctgα	ctgα	tgα	-tgα	-ctgα	ctgα
Function (angle in rad.)	π/2 – α	π/2 + α	π – α	π + α	3π/2 – α	3π/2 + α	2π-α	2π + α

Parity of trigonometric functions. Angles φ and -φ are formed by turning the beam in two mutually opposite directions (clockwise and counterclockwise).
Therefore, the end sides OA 1 and OA 2 of these angles are symmetrical about the x-axis. Unit vector coordinates OA 1 = ( X 1 , at 1) and ОА 2 = ( X 2 , y 2) satisfy the relations: X 2 = X 1 y 2 = -at 1 Therefore cos(-φ) = cosφ, sin (- φ) = -sin φ, Therefore, the sine is an odd function, and the cosine is an even function of the angle.
Next we have:
therefore tangent and cotangent are odd functions of an angle.

8)Inverse trigonometric functions- mathematical functions that are inverse to trigonometric functions. Inverse trigonometric functions usually include six functions:

§ arcsine(symbol: arcsin)

§ arc cosine(symbol: arccos)

§ arc tangent(designation: arctg; in foreign literature arctan)

§ arc tangent(designation: arcctg; in foreign literature arccotan)

§ arcsecant(symbol: arcsec)

§ arccosecant(designation: arccosec; in foreign literature arccsc)

The name of the inverse trigonometric function is formed from the name of the corresponding trigonometric function by adding the prefix "ark-" (from lat. arc- arc). This is due to the fact that geometrically the value of the inverse trigonometric function can be associated with the length of the arc of a unit circle (or the angle that subtends this arc) corresponding to one or another segment. Occasionally in foreign literature they use designations like sin −1 for the arcsine, etc.; this is considered unjustified, since confusion with raising the function to the power of −1 is possible.

properties of the arcsin function

(function is odd). at .

at

at

Properties of function arccos[

· (the function is centrally symmetric with respect to the point ), is indifferent.

·

·

·

arctg function properties

·

· , for x > 0.

arcctg function properties

(the graph of the function is centrally symmetric with respect to the point

· for any

·

12) The power of the number a > 0 with a rational exponent is the exponent, the exponent of which can be represented as an ordinary irreducible fraction x = m / n, where m is an integer and n is a natural number, and n > 1 (x is the exponent).

Degree with real exponent

Let a positive number and an arbitrary real number be given. The number is called the degree, the number is the base of the degree, the number is the exponent.

By definition it is assumed:

If and are positive numbers, and are any real numbers, then the following properties are true:

14)Base logarithm of a number(from the Greek λόγος - "word", "relation" and ἀριθμός - "number") is defined as an indicator of the degree to which the base must be raised to get a number. Designation:, pronounced: " base logarithm".

Properties of logarithms:

1° - basic logarithmic identity.

The logarithm of unity in any positive base other than 1 is zero. This is possible because from any real number you can get 1 only by raising it to the zero power.

4° is the logarithm of the product.

The logarithm of the product is equal to the sum of the logarithms of the factors.

5° is the logarithm of the quotient.

The logarithm of the quotient (fraction) is equal to the difference of the logarithms of the factors.

6° is the logarithm of the degree.

The logarithm of a degree is equal to the product of the exponent and the logarithm of its base.

7°

8°

9° - transition to a new basis.

15) Real number - (real number), any positive, negative number or zero. By means of real numbers, the results of measurement of all physical quantities are expressed. ;

16)imaginary unit is usually a complex number whose square is equal to negative one. However, other options are also possible: in the construction of doubling according to Cayley-Dixon or in the framework of algebra according to Clifford.

Complex numbers(obsolete imaginary numbers) - numbers of the form , where and are real numbers, is an imaginary unit; that is . The set of all complex numbers is usually denoted from lat. complex- closely related.

Lesson topic: Degree with rational and real exponents.

Goals:

Educational :

• generalize the concept of degree;

  to develop the ability to find the value of the degree with a real indicator;

  consolidate the ability to use the properties of the degree when simplifying expressions;

  develop the skill of using the properties of the degree in calculations.

Educational :

• intellectual, emotional, personal development of the student;

  develop the ability to generalize, systematize on the basis of comparison, draw a conclusion;

  activate independent activity;

  develop curiosity.

Educational :

• education of communicative and informational culture of students;

  aesthetic education is carried out through the formation of the ability to rationally, accurately draw up a task on a blackboard and in a notebook.

Students should know: definition and properties of degree with real exponent

Students should be able to:

determine whether an expression with a degree makes sense;

use the properties of the degree in calculations and simplification of expressions;

solve examples containing a degree;

compare, find similarities and differences.

Lesson form: seminar - workshop, with elements of research. Computer support.

Form of organization of training: individual, group.

Pedagogical technologies Keywords: problem-based learning, learning in cooperation, personality-oriented learning, communicative.

Lesson type: lesson of research and practical work.

Lesson visuals and handouts:

presentation

formulas and tables (application 1.2)

assignment for independent work (Appendix 3)

Lesson plan

№

Lesson stage

Purpose of the stage

Time, min.

Lesson start

Reporting the topic of the lesson, setting the goals of the lesson.

1-2 min

oral work

Review the power formulas.

Degree properties.

4-5 min.

Frontal solution

boards from textbook No. 57 (1,3,5)

№58(1,3,5) with detailed adherence to the solution plan.

Formation of skills and abilities

have students apply properties

degrees when finding the values ​​of the expression.

8-10 min.

Work in microgroups.

Identifying gaps in knowledge

students, creating conditions for

student's individual development

at the lesson.

15-20 min.

Summing up the work.

Track the success of your work

Students, when independently solving problems on a topic, find out

the nature of the difficulties, their causes,

provide collective solutions.

5-6 min.

Homework

Introduce students to homework. Give the necessary explanations.

1-2 min.

DURING THE CLASSES

Organizing time

Hello guys! Write in your notebooks the number, the topic of the lesson.

They say that the inventor of chess, as a reward for his invention, asked the Raja for some rice: on the first cell of the board he asked to put one grain, on the second - 2 times more, i.e. 2 grains, on the third - 2 more times more, i.e. 4 grains, etc. up to 64 cells.

His request seemed too modest to the Raja, but it soon became clear that it was impossible to fulfill it. The number of grains that had to be given to the inventor of chess as a reward is expressed by the sum

1+2+2 2 +2 3 +…+2 63 .

This amount is equal to a huge number

18446744073709551615

And it is so large that this amount of grain could cover the entire surface of our planet, including the world ocean, with a layer of 1 cm.

Degrees are used when writing numbers and expressions, which makes them more compact and convenient for performing actions.

Often, degrees are used when measuring physical quantities, which can be "very large" and "very small".

The mass of the Earth 6000000000000000000000t is written as a product of 6.10 21 t

The diameter of a water molecule 0.0000000003 m is written as a product

3.10 -10 m.

1. What mathematical concept is associated with the words:

Base
Index(Degree)

What words can combine the words:
rational number
Integer
Natural number
irrational number(Real number)
Formulate the topic of the lesson.(Power with real exponent)

2. So a x,wherex is a real number. Select from expressions

With natural indicator

With an integer

with rational exponent

With an irrational exponent

3. What is our goal?(USE)
What kindobjectives of our lesson ?
- Generalize the concept of degree.

Tasks:

– repeat the properties of the degree
– consider the use of degree properties in calculations and simplifications of expressions
– development of computational skills

4 . Degree with rational exponent

Base

degrees

Degree with exponentr, base a (nN, mn

r= n

r= - n

r= 0

r= 0

r=0

a n= a. a. … . a

a -n=

a 0 =1

a n=a.a. ….a

a -n=

Does not exist

Does not exist

a 0 =1

a=0

0 n=0

Does not exist

Does not exist

Does not exist

5 . From these expressions, choose those that do not make sense:

6 . Definition

If numberr- natural, then rthere is a workrnumbers, each of which is equal to a:

a r= a. a. … . a

If numberr- fractional and positive, that is, wheremandn- natural

numbers, then

If the indicatorris rational and negative, then the expressiona r

is defined as the reciprocal ofa - r

or

If

7 . For example

8 . Powers of positive numbers have the following basic properties:

9 . Calculate

10. What actions (mathematical operations) can be performed with degrees?

Set match:

A) When multiplying powers with equal bases

1) The bases are multiplied, but the exponent remains the same

B) When dividing degrees with equal bases

2) The bases are divided, but the exponent remains the same

B) When raising a power to a power

3) The base remains the same, but the exponents are multiplied

D) When multiplying powers with equal exponents

4) The base remains the same, and the exponents are subtracted

E) When dividing degrees with equal indicators

5) The base remains the same, and the indicators add up

11 . From the textbook (at the blackboard)

For a class solution:

№57 (1,3,5)

№58 (1, 3, 5)

№59 (1, 3)

№60 (1,3)

12 . According to the materials of the exam

(independent work) on leaflets

XIVcentury.

Answer: Oresma. 13. Additionally (individually) for those who can complete the tasks faster:

14. Homework

§ 5 (know definitions, formulas)

№57 (2, 4, 6)

№58 (2,4)

№59 (2,4)

№60 (2,4) .

At the end of the lesson:

“Mathematics already then needs to be taught, that it puts the mind in order”

– So said the great Russian mathematician Mikhail Lomonosov.

- Thank you for the lesson!

Appendix 1

1. Degrees. Basic properties

indicator

a 1 =a

a n=a.a. ….a

aRn

3 5 =3 . 3 . 3 . 3 . 3 . 3=243,

(-2) 3 =(-2) . (-2) . (-2)= - 8

Degree with integer exponent

a 0 =1,

where a

0 0 - not defined.

Degree with rational

indicator

wherea

m n

Degree with irrational exponent

Answer: ==25.9...

1. a x. a y=a x+y

2.a x: a y==a x-y

3. .(a x) y=a x.y

4.(a.b) n=a n.b n

5. (=

6. (

Annex 2

2. Degree with a rational exponent

Base

degrees

Degree with exponentr, base a (nN, mn

r= n

r= - n

r= 0

r= 0

r=0

a n= a. a. … . a

a -n=

a 0 =1

a n=a.a. ….a

a -n=

Does not exist

Does not exist

a 0 =1

a=0

0 n=0

Does not exist

Does not exist

Does not exist

Annex 3

3. Independent work

For the first time, actions on powers were used by a French mathematicianXIVcentury.

Decipher the name of the French scientist.