Basic information about monomials contains the clarification that any monomial can be reduced to a standard form. In the material below we will consider this issue in more detail: we will outline the meaning of this action, we will define the steps that allow us to set the standard form of a monomial, and also consolidate the theory by solving examples.
The meaning of reducing a monomial to standard form
Writing a monomial in standard form makes it more convenient to work with it. Often monomials are specified in a non-standard form, and then it becomes necessary to carry out identical transformations to bring the given monomial into a standard form.
Definition 1
Reducing a monomial to standard form is the performance of appropriate actions (identical transformations) with a monomial in order to write it in standard form.
Method for reducing a monomial to standard form
From the definition it follows that a monomial of a non-standard form is a product of numbers, variables and their powers, and their repetition is possible. In turn, a monomial of the standard type contains in its notation only one number and non-repeating variables or their powers.
To bring a non-standard monomial into standard form, you must use the following rule for reducing a monomial to standard form:
- the first step is to group numerical factors, identical variables and their powers;
- the second step is to calculate the products of numbers and apply the property of powers with equal bases.
Examples and their solutions
Example 1Given a monomial 3 x 2 x 2 . It is necessary to bring it to a standard form.
Solution
Let us group numerical factors and factors with variable x, as a result the given monomial will take the form: (3 2) (x x 2) .
The product in parentheses is 6. Applying the rule of multiplication of powers with the same bases, we present the expression in brackets as: x 1 + 2 = x 3. As a result, we obtain a monomial of the standard form: 6 x 3.
A short version of the solution looks like this: 3 · x · 2 · x 2 = (3 · 2) · (x · x 2) = 6 · x 3 .
Answer: 3 x 2 x 2 = 6 x 3.
Example 2
The monomial is given: a 5 · b 2 · a · m · (- 1) · a 2 · b . It is necessary to bring it into a standard form and indicate its coefficient.
Solution
the given monomial has one numerical factor in its notation: - 1, let’s move it to the beginning. Then we will group the factors with the variable a and the factors with the variable b. There is nothing to group the variable m with, so we leave it in its original form. As a result of the above actions we get: - 1 · a 5 · a · a 2 · b 2 · b · m.
Let's perform operations with powers in brackets, then the monomial will take the standard form: (- 1) · a 5 + 1 + 2 · b 2 + 1 · m = (- 1) · a 8 · b 3 · m. From this entry we can easily determine the coefficient of the monomial: it is equal to - 1. It is quite possible to replace minus one simply with a minus sign: (- 1) · a 8 · b 3 · m = - a 8 · b 3 · m.
A short record of all actions looks like this:
a 5 b 2 a m (- 1) a 2 b = (- 1) (a 5 a a 2) (b 2 b) m = = (- 1) a 5 + 1 + 2 b 2 + 1 m = (- 1) a 8 b 3 m = - a 8 b 3 m
Answer:
a 5 · b 2 · a · m · (- 1) · a 2 · b = - a 8 · b 3 · m, the coefficient of the given monomial is - 1.
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The concept of a monomial
Definition of a monomial: A monomial is an algebraic expression that uses only multiplication.
Standard form of monomial
What is the standard form of a monomial? A monomial is written in standard form, if it has a numerical factor in the first place and this factor is called the coefficient of the monomial, there is only one in the monomial, the letters of the monomial are arranged in alphabetical order and each letter appears only once.
An example of a monomial in standard form:
here in the first place is the number, the coefficient of the monomial, and this number is only one in our monomial, each letter occurs only once and the letters are arranged in alphabetical order, in this case it is the Latin alphabet.
Another example of a monomial in standard form:
each letter occurs only once, they are arranged in Latin alphabetical order, but where is the coefficient of the monomial, i.e. the numeric factor that should come first? Here it is equal to one: 1adm.
Can the coefficient of a monomial be negative? Yes, maybe, example: -5a.
Can the coefficient of a monomial be fractional? Yes, maybe, example: 5.2a.
If a monomial consists only of a number, i.e. has no letters, how can I bring it to standard form? Any monomial that is a number is already in standard form, for example: the number 5 is a monomial in standard form.
Reducing monomials to standard form
How to bring a monomial to standard form? Let's look at examples.
Let the monomial 2a4b be given; we need to bring it to standard form. We multiply its two numerical factors and get 8ab. Now the monomial is written in standard form, i.e. has only one numerical factor, written in the first place, each letter in the monomial occurs only once and these letters are arranged in alphabetical order. So 2a4b = 8ab.
Given: monomial 2a4a, bring the monomial to standard form. We multiply the numbers 2 and 4, replacing the product aa with the second power of a 2. We get: 8a 2 . This is the standard form of this monomial. So 2a4a = 8a 2 .
Similar monomials
What are similar monomials? If monomials differ only in coefficients or are equal, then they are called similar.
Example of similar monomials: 5a and 2a. These monomials differ only in coefficients, which means they are similar.
Are the monomials 5abc and 10cba similar? Let's bring the second monomial to standard form and get 10abc. Now we can see that the monomials 5abc and 10abc differ only in their coefficients, which means that they are similar.
Addition of monomials
What is the sum of the monomials? We can only sum similar monomials. Let's look at an example of adding monomials. What is the sum of the monomials 5a and 2a? The sum of these monomials will be a monomial similar to them, the coefficient of which is equal to the sum of the coefficients of the terms. So, the sum of the monomials is 5a + 2a = 7a.
More examples of adding monomials:
2a 2 + 3a 2 = 5a 2
2a 2 b 3 c 4 + 3a 2 b 3 c 4 = 5a 2 b 3 c 4
Again. You can only add similar monomials; addition comes down to adding their coefficients.
Subtracting monomials
What is the difference between the monomials? We can only subtract similar monomials. Let's look at an example of subtracting monomials. What is the difference between monomials 5a and 2a? The difference of these monomials will be a monomial similar to them, the coefficient of which is equal to the difference of the coefficients of these monomials. So, the difference of the monomials is 5a - 2a = 3a.
More examples of subtracting monomials:
10a 2 - 3a 2 = 7a 2
5a 2 b 3 c 4 - 3a 2 b 3 c 4 = 2a 2 b 3 c 4
Multiplying monomials
What is the product of monomials? Let's look at an example:
those. the product of monomials is equal to a monomial whose factors are made up of the factors of the original monomials.
Another example:
2a 2 b 3 * a 5 b 9 = 2a 7 b 12 .
How did this result come about? Each factor contains “a” to the power: in the first - “a” to the power of 2, and in the second - “a” to the power of 5. This means that the product will contain “a” to the power of 7, because when multiplying identical letters, the exponents of their powers fold up:
A 2 * a 5 = a 7 .
The same applies to the factor “b”.
The coefficient of the first factor is two, and the second is one, so the result is 2 * 1 = 2.
This is how the result was calculated: 2a 7 b 12.
From these examples it is clear that the coefficients of monomials are multiplied, and identical letters are replaced by the sums of their powers in the product.
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
Lesson on the topic: "Standard form of a monomial. Definition. Examples"
Additional materials
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Teaching aids and simulators in the Integral online store for grade 7
Electronic textbook "Understandable Geometry" for grades 7-9
Multimedia textbook "Geometry in 10 minutes" for grades 7-9
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 natural indicator:
42;
3;
0;
Standard form of monomial
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.
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:
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: $\frac(4)(5)*a^3$.
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.
