Lesson on the topic: "Standard form of a monomial. Definition. Examples"
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Monomial. Definition
Monomial is a mathematical expression that is the product of a prime factor and one or more variables.Monomials include all numbers, variables, their powers with a natural exponent:
42; 3; 0; 6 2 ; 2 3 ; b 3 ; ax 4 ; 4x 3 ; 5a 2 ; 12xyz 3 .
Quite often it is difficult to determine whether a given mathematical expression refers to a monomial or not. For example, $\frac(4a^3)(5)$. Is this a monomial or not? To answer this question we need to simplify the expression, i.e. present in the form: $\frac(4)(5)*a^3$.
We can say for sure that this expression is a monomial.
Standard form of monomial
When performing calculations, it is advisable to reduce the monomial to standard form. This is the most concise and understandable recording of a monomial.The procedure for reducing a monomial to standard form is as follows:
1. Multiply the coefficients of the monomial (or numerical factors) and place the resulting result in first place.
2. Select all powers with the same letter base and multiply them.
3. Repeat point 2 for all variables.
Examples.
I. Reduce the given monomial $3x^2zy^3*5y^2z^4$ to standard form.
Solution.
1. Multiply the coefficients of the monomial $15x^2y^3z * y^2z^4$.
2. Now we present similar terms $15x^2y^5z^5$.
II. Reduce the given monomial $5a^2b^3 * \frac(2)(7)a^3b^2c$ to standard form.
Solution.
1. Multiply the coefficients of the monomial $\frac(10)(7)a^2b^3*a^3b^2c$.
2. Now we present similar terms $\frac(10)(7)a^5b^5c$.
In this lesson we will give a strict definition of a monomial and look at various examples from the textbook. Let us recall the rules for multiplying powers with the same bases. Let us define the standard form of a monomial, the coefficient of the monomial and its letter part. Let's consider two main typical operations on monomials, namely reduction to a standard form and calculation of a specific numerical value of a monomial for given values of the literal variables included in it. Let us formulate a rule for reducing a monomial to standard form. Let's learn how to solve standard problems with any monomials.
Subject:Monomials. Arithmetic operations on monomials
Lesson:The concept of a monomial. Standard form of monomial
Consider some examples:
3. 
;
Let us find common features for the given expressions. In all three cases, the expression is the product of numbers and variables raised to a power. Based on this we give monomial definition : A monomial is an algebraic expression that consists of the product of powers and numbers.
Now we give examples of expressions that are not monomials:
Let us find the difference between these expressions and the previous ones. It consists in the fact that in examples 4-7 there are addition, subtraction or division operations, while in examples 1-3, which are monomials, there are no these operations.
Here are a few more examples:
Expression number 8 is a monomial because it is the product of a power and a number, whereas example 9 is not a monomial.
Now let's find out actions on monomials .
1. Simplification. Let's look at example No. 3 ;and example No. 2 /
In the second example we see only one coefficient - , each variable occurs only once, that is, the variable " A" is represented in a single copy as "", similarly, the variables "" and "" appear only once.
In example No. 3, on the contrary, there are two different coefficients - and , we see the variable "" twice - as "" and as "", similarly, the variable "" appears twice. That is, this expression should be simplified, thus we arrive at the first action performed on monomials is to reduce the monomial to standard form . To do this, we will reduce the expression from Example 3 to standard form, then we will define this operation and learn how to reduce any monomial to standard form.
So, consider an example:
The first action in the operation of reduction to standard form is always to multiply all numerical factors:
;
The result of this action will be called coefficient of the monomial .
Next you need to multiply the powers. Let's multiply the powers of the variable " X"according to the rule for multiplying powers with the same bases, which states that when multiplying, the exponents are added:
Now let's multiply the powers " at»:
;
So, here is a simplified expression:
;
Any monomial can be reduced to standard form. Let's formulate standardization rule :
Multiply all numerical factors;
Place the resulting coefficient in first place;
Multiply all degrees, that is, get the letter part;
That is, any monomial is characterized by a coefficient and a letter part. Looking ahead, we note that monomials that have the same letter part are called similar.
Now we need to work out technique for reducing monomials to standard form . Consider examples from the textbook:
Assignment: bring the monomial to standard form, name the coefficient and the letter part.
To complete the task, we will use the rule for reducing a monomial to a standard form and the properties of powers.
1. 
;
3. 
;
Comments on the first example: First, let's determine whether this expression is really a monomial; to do this, let's check whether it contains operations of multiplication of numbers and powers and whether it contains operations of addition, subtraction or division. We can say that this expression is a monomial since the above condition is satisfied. Next, according to the rule for reducing a monomial to a standard form, we multiply the numerical factors:
- we found the coefficient of a given monomial;
; ; ; that is, the literal part of the expression is obtained:;
Let's write down the answer: ;
Comments on the second example: Following the rule we perform:
1) multiply numerical factors:
2) multiply the powers:
Variables are presented in a single copy, that is, they cannot be multiplied with anything, they are rewritten without changes, the degree is multiplied:
Let's write down the answer:
;
In this example, the coefficient of the monomial is equal to one, and the letter part is .
Comments on the third example: a Similar to the previous examples, we perform the following actions:
1) multiply numerical factors:
;
2) multiply the powers:
;
Let's write down the answer: ;
In this case, the coefficient of the monomial is “”, and the letter part 
.
Now let's consider second standard operation on monomials . Since a monomial is an algebraic expression consisting of literal variables that can take on specific numeric values, we have an arithmetic numeric expression that must be evaluated. That is, the next operation on polynomials is calculating their specific numerical value .
Let's look at an example. Monomial given:
this monomial has already been reduced to standard form, its coefficient is equal to one, and the letter part
Earlier we said that an algebraic expression cannot always be calculated, that is, the variables that are included in it cannot take on any value. In the case of a monomial, the variables included in it can be any; this is a feature of the monomial.
So, in the given example, you need to calculate the value of the monomial at , , , .
I. Expressions that are made up of numbers, variables and their powers using the action of multiplication are called monomials.
Examples of monomials:
A) a; b) ab; V) 12; G)-3c; d) 2a 2 ∙(-3.5b) 3 ; e)-123.45xy 5 z; and) 8ac∙2.5a 2 ∙(-3c 3).
II. This type of monomial, when the numerical factor (coefficient) comes first, followed by the variables with their powers, is called the standard type of monomial.
Thus, the monomials given above, under the letters a B C), G) And e) written in standard form, and the monomials under the letters d) And and) it is required to bring it to a standard form, i.e. to a form where the numerical factor comes first, followed by the letter factors with their exponents, and the letter factors are in alphabetical order. Let us present monomials d) And and) to the standard view.
d) 2a 2 ∙(-3.5b) 3=2a 2 ∙(-3.5) 3 ∙b 3 =-2a 2 ∙3.5∙3.5∙3.5∙b 3 = -85.75a 2 b 3 ;
and) 8ac∙2.5a 2 ∙(-3c 3)=-8∙2.5∙3a 3 c 3 = -60a 3 c 3 .
III.The sum of the exponents of all variables included in a monomial is called the degree of the monomial.
Examples. What degree do monomials have? a) - g)?
a) a. First;
b) ab. Second: A in the first degree and b to the first power - the sum of indicators 1+1=2 ;
V) 12. Zero, since there are no letter factors;
G) -3c. First;
d) -85.75a 2 b 3 . Fifth. We have reduced this monomial to standard form, we have A to the second degree and b in the third. Let's add up the indicators: 2+3=5 ;
e) -123.45xy 5 z. Seventh. We added up the exponents of the letter factors: 1+5+1=7 ;
and) -60a 3 c 3 . Sixth, since the sum of the exponents of the letter factors 3+3=6 .
IV. Monomials that have the same letter part are called similar monomials.
Example. Indicate similar monomials among the given monomials 1) -7).
1) 3aabbc; 2) -4.1a 3 bc; 3) 56a 2 b 2 c; 4) 98.7a 2 bac; 5) 10aaa 2 x; 6) -2.3a 4 x; 7) 34x 2 y.
Let us present monomials 1), 4) And 5) to the standard view. Then the line of monomials data will look like this:
1) 3a 2 b 2 c; 2) -4.1a 3 bc; 3) 56a 2 b 2 c; 4) 98.7a 3 bc; 5) 10a 4x; 6) -2.3a 4 x; 7) 34x 2 y.
Similar will be those that have the same letter part, i.e. 1) and 3) ; 2) and 4); 5) and 6).
1) 3a 2 b 2 c and 3) 56a 2 b 2 c;
2) -4.1a 3 bc and 4) 98.7a 3 bc;
5) 10a 4 x and 6) -2.3a 4 x.
Goal: -To become familiar with the concept of a monomial;
Develop the ability to give examples of monomials
Determine whether an expression is a monomial
Indicate its coefficient and letter part.
Get acquainted with the concept of “standard form of monomial”
Enter an algorithm for reducing a monomial to a standard form;
Develop practical skills in using the algorithm
bringing a monomial to standard form.
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TOPIC: The concept of a monomial. Standard form of a monomial Purpose: -To become familiar with the concept of a monomial; -Develop the ability to give examples of monomials; -Determine whether an expression is a monomial; -Indicate its coefficient and letter part. -Get acquainted with the concept of “standard form of a monomial” -Introduce an algorithm for bringing a monomial to a standard form; Develop practical skills in applying the algorithm for reducing a monomial to a standard form.
A SINGLE TERM IS AN ALGEBRAIC EXPRESSION WHICH IS THE PRODUCT OF NUMBERS AND VARIABLES RAISED TO A POWER WITH A NATURAL EXponent. 2av, - 4а⁴в⁵, 1.7с⁸в⁴ 0; 2 ; -0.6; X; A; x ⁶ Are not a monomial of an expression of the form: a+b; 2x⁴+ 3y⁹; а⁴⁄с ⁸ CONCEPT OF MONOMIAL
Consider the monomial: 3a∙4 a²b⁵c²bac⁵=3∙4aa²b⁵bc²c=12a³b⁶c³ Mathematics strives for clarity, brevity and order. We have reduced the monomial to a shorter notation i.e. to the standard view.
Algorithm. Reduce the monomial to standard form and name the coefficient of the monomial. 3x⁴ yz ∙(-2) xy⁴z ⁸= 3∙(- 2) x⁴∙ x ∙ y⁴∙ y∙z∙z ⁸ = = -6x⁵∙ y⁵∙z ⁹ ¼ab⁴c4c=¼∙4ab⁴(c∙c )=ab⁴c² ( 3 /10) av To bring a monomial to standard form, you need to: 1) Multiply all numerical factors and put their product in first place; 2) Multiply all available powers with the same letter base; 3) Multiply all available powers with another letter base, etc. The numerical factor of a monomial written in standard form is called the coefficient of the monomial
Reduce the monomial to standard form. Option 1 a) 7с⁴·4с³·8 c⁶ b) 8х²·4 y³·(- 2х ³) Option 2 a) 6 n²·3n³·9n⁶ b) 15 q⁴·2p²·(-5p⁵)
Let's check the answers for independent work. Option 1 a) 244 s¹³ b) -64 x ⁸ y³ Option 2 a) 162 n ¹¹ b) - 150 q ⁴ p⁷
On the topic: methodological developments, presentations and notes
Presentation in mathematics on the topic "The concept of a monomial. The standard form of a monomial." The presentation was compiled to consider a new topic in mathematics in grade 7 "The concept of a monomial. The standard form of a monomial...
concept of monomial. standard form of monomial
presentation for an algebra lesson in 7th grade on the topic "The concept of a monomial. The standard form of a monomial." The concepts of monomial, degree of monomial, coefficient of monomial, standard form of monomial are given....
Monomials are products of numbers, variables and their powers. Numbers, variables and their powers are also considered monomials. For example: 12ac, -33, a^2b, a, c^9. The monomial 5aa2b2b can be reduced to the form 20a^2b^2. This form is called the standard form of the monomial. That is, the standard form of the monomial is the product of the coefficient (which comes first) and the powers of the variables. Coefficients 1 and -1 are not written, but a minus is kept from -1. Monomial and its standard form
The expressions 5a2x, 2a3(-3)x2, b2x are products of numbers, variables and their powers. Such expressions are called monomials. Numbers, variables and their powers are also considered monomials.
For example, the expressions 8, 35,y and y2 are monomials.
The standard form of a monomial is a monomial in the form of the product of a numerical factor in first place and powers of various variables. Any monomial can be reduced to a standard form by multiplying all the variables and numbers included in it. Here is an example of reducing a monomial to standard form:
4x2y4(-5)yx3 = 4(-5)x2x3y4y = -20x5y5
The numerical factor of a monomial written in standard form is called the coefficient of the monomial. For example, the coefficient of the monomial -7x2y2 is equal to -7. The coefficients of the monomials x3 and -xy are considered equal to 1 and -1, since x3 = 1x3 and -xy = -1xy
The degree of a monomial is the sum of the exponents of all the variables included in it. If a monomial does not contain variables, that is, it is a number, then its degree is considered equal to zero.
For example, the degree of the monomial 8x3yz2 is 6, the monomial 6x is 1, and the degree of -10 is 0.
Multiplying monomials. Raising monomials to powers
When multiplying monomials and raising monomials to a power, the rule for multiplying powers with the same base and the rule for raising a power to a power are used. This produces a monomial, which is usually represented in standard form.
For example
4x3y2(-3)x2y = 4(-3)x3x2y2y = -12x5y3
((-5)x3y2)3 = (-5)3x3*3y2*3 = -125x9y6