Multiplying a Monomial by a Monomial: Multiplication is one of the fundamental mathematical operations used in algebraic expressions. We can categorise algebraic expressions depending on the number of terms they include, such as monomial, binomial, trinomial, quadrinomial, or polynomial. A monomial expression is an algebraic expression that contains only one term, which comprises a variable and its coefficients.

Monomial multiplication is one of the simplest operations in mathematics. Using certain predefined principles, we can easily multiply one monomial by another. This article will explain what monomials are, how to multiply monomials, the many types of monomials, the rules for multiplying monomials, and how to multiply monomials.

Algebraic Expression

The combination of constants and variables connected by any or all of the four arithmetic operations such as addition \(\left( + \right)\), subtraction \(\left( – \right),\) multiplication \(\left( \times \right)\), and division \(\left( \div \right)\) is known as an algebraic expression.

For example, \(x + 7,\,a – 5,\,4y + 5,\,3y,\,6{a^2}b\) etc.

Constant

Any quantity whose value never changes or whose value is fixed is called a constant.

Example: \(8,\,6,\,\pi ,\,7 \ldots \ldots ,\) etc. are constants

Variable

Any quantity whose value changes and varies is called a variable.

Example: Conventionally, alphabets like \(a,\,b,\,p,\,q,\,x,\,y,\,z,\, \ldots .\) are used as variables.

Terms

In an algebraic expression, a term may consist of only constant, only one variable, the product of two or more variables, a product of both the variable and the constant part. The terms may be positive or negative. An algebraic expression may contain one term or more than one term.

For example, \(2,\, – 1,\, – x,\,y,\,xy,\,yz,\,a{b^2}\) etc. are the terms.

Coefficients

In an algebraic expression with the product of numbers and literals, any of the factors is called the coefficient of the product of the other factors.

Example: In the term \(5xyz\), the numeral coefficient is \(5\), coefficient of \(x\) is \(5yz\), coefficient of \(y\) is \(5xz\) etc.

Degree

The degree of a polynomial is the highest power of the variables of its terms when a polynomial is expressed in its general form. It is the sum of exponents of the variables in the term if it has more than one variable.

For example: In \({x^2}\), the degree of \({x^2}\) is \(2\).

Monomial

The algebraic expressions with the exponents of the variables as whole numbers are polynomials. Polynomials with only one term are called monomials. They are algebraic expressions that contain only one term.
Example: \(5x,\,2xy,\, – 3{a^2}b,\, – 7\) etc., are monomials.

Parts of a Monomial

A monomial has few parts. These are:

(i) Variable: The letters or the alphabets present in the monomial expression.
(ii) Degree: The sum of the exponents of the variable/variables present in the expression.
(iii) Coefficient: The constant number which is multiplied by the variable/variables in the expression

For example, in \(8{x^2},\,8\) is the coefficient and \(x\) is the variable and \(2\) is the exponent

Types of Monomials

We can categorise monomial into two types such as constant monomial and variable monomial.

Constant Monomial

It is a monomial with a constant term, or we can represent it by any number. It can be a negative number, a positive number and can be a fraction. It does not consist of variables. The constant monomials are \(2,\,5,\, – 7,\,\frac{5}{6}\), etc.

Variable Monomial

It is a monomial with a variable or the variables, or we can express it by letters such as \(a,\,b,\,c,\,x,\,y,\,z\) etc.
Variable monomials may contain only or more variables. \(2a,\,6{x^2},\, – 7xy,\,abc\), etc., are the variable monomials.
Here, \(2a\) and \(6{x^2}\) are the monomials with a single variable, and \( – 7xy\) is a monomial with two variables, and \(abc\) is a monomial with three variables.

Multiplication of Monomials

Monomial multiplication is a method for multiplying two or more monomials at a time. The multiplication of a monomial by another monomial gives a monomial as a product. Based on the types of polynomials we use, there are different ways of multiplying them.

There are some specified rules of multiplication for different types of monomials. The constant-coefficient is multiplied with a constant coefficient, and the variable is multiplied with a variable.

Multiplying a Constant Monomial With a Variable Monomial

Let us consider two monomials \(7\) and \(6y\). In this case, \(7\) is a constant monomial, and \(6y\) is a variable monomial. We will multiply the constant monomial with the coefficient of the variable monomial. It gives \(7 \times 6 = 42\). After that, we will write the variable \((y)\) after \(42\).

Hence, the answer is
\(7 \times 6y = 42y\)

Multiplying Two Monomials With Different Variables

Considering two monomials with different variables, \(2{x^3}\) & \(5y\)

1. First, we will multiply the coefficients. The coefficient of \(2{x^3}\) is \(2\) and the coefficient of \(5y\) is \(5\). After multiplying, we get \(2 \times 5 = 10\)
2. Next, we will multiply the variables using the rule of the exponent wherever it is required. Here, the variable parts are \({x^3}\) & \(y\). Multiplying these we get, \({x^3} \times y = {x^3}y\) as the variables are different. We can multiply them without using the exponent rule.
3. Hence, the answer is
   \(2{x^3} \times 5y = 10{x^3}y\)

Multiplying Two Monomials With Same Variable

Let us learn the following steps using the example given below.
Considering two monomials \(4{a^2}\) & \(3{a^4}\)

1. First, we will multiply the coefficients. The coefficient of \(4{a^2}\) is \(4\) and the coefficient of \(3{a^4}\) is \(3\). After multiplying, we get \(3 \times 4 = 12\)
2. Next, we will multiply the variables using the rule of the exponents. Here, the variable parts are \({a^2}\) & \({a^4}\). Multiplying these we get, \({a^2} \times {a^4} = {a^6}\) as we added the exponents of the variable as per the rule of the exponent.
3. Hence, the answer is
   \(4{a^2} \times 3{a^4} = 12{a^6}\)

Solved Examples on Multiplying a Monomial by a Monomial

Q.1. Multiply \(x\) and \({x^2}\)
Ans: Given two monomials are \(x\) & \({x^2}\)
1. First, we will multiply the coefficients. The coefficient of both monomials is \(1\). So, the product is \(1\).
2. Next, we will multiply the variables using the rule of the exponents. Here, the variable parts are \(x\) & \({x^2}\). Multiplying these, we get, \(x \times {x^2} = {x^3}\) as we added the exponents of the variable as per the rule of the exponent.3. Hence, the answer is \(x \times {x^2} = {x^3}\).
3. Hence, the answer is  \(x \times {x^2} = {x^3}\).

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Q.2. Multiply \(8\) and \(6{y^3}\)
Ans: Given two monomials are \(8\) & \(6{y^3}\).
In this case, \(8\) is a constant monomial, and \(6{y^3}\) is a variable polynomial. We will multiply the constant monomial with the coefficient of the variable polynomial. It gives \(8 \times 7 = 56\). After that, we will write the variable part \(\left( {{y^3}} \right)\) after \(56\).
Hence, the answer is \(8 \times 6{y^3} = 56{y^3}\).

Q.3. Multiply \(3x\) and \(4y\).
Ans: Given two monomials are \(3x\) & \(4y\)
1. First, we will multiply the coefficients. The coefficient of \(3x\) is \(3\) and the coefficient of \(4y\) is \(4\). After multiplying, we get \(3 \times 4 = 12\)
2. Next, we will multiply the variables using the rule of the exponents wherever it is required. Here, the variable parts are \(x,\,y\). Multiplying these, we get, \(x \times y = xy\). As the variables are different, we can multiply them without using the exponent rule.
3. Hence, the answer is \(3x \times 4y = 12xy\).

Q.4. Multiply \(2abc\) and \({a^2}b\)
Ans: Given two monomials are \(2abc\) & \({a^2}b\)
\(2abc\) is a monomial with three variables and \({a^2}b\) is a monomial with two variables.
1. First, we will multiply the coefficients. We get \(2 \times 1 = 2\)
2. Next, we will multiply the variables using the rule of the exponents wherever it is required. Here, the variable parts are \(abc\) & \({a^2}b\). Multiplying these, we get, \(abc \times {a^2}b = \left( {a \times {a^2}} \right) \times (b \times b) \times c = {a^3}{b^2}c\) (as we added the exponents of the same variables as per the exponent rule.)
3. Hence, the answer is \(2abc \times {a^2}b = 2{a^3}{b^2}c\).

Q.5. Multiply \(7{z^3}\) and \(9{z^2}\)
Ans: Given two monomials are \(7{z^3}\) & \(9{z^2}\)
1. First, we will multiply the coefficients. The coefficient of \(7{z^3}\) is \(7\) and the coefficient of \(9{z^2}\) is \(9\). After multiplying, we get \(7 \times 9 = 63\)
2. Next, we will multiply the variables using the rule of the exponent. Here, the variable parts are \({z^3},\,{z^2}\). Multiplying these we get, \({z^3} \times {z^2} = {z^5}\) as we added the exponents of the variable as per the rule of the exponent.
3. Hence, the answer is \(63{z^5}\).

Summary

In this article, we learnt about monomial, types of monomial, parts of monomial, multiplying two monomials. We also discussed the method of multiplication of two monomials with different variables and the same variable. At last, we solved some examples of the multiplication of two monomials.

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Frequently Asked Questions (FAQs)

Q.1. How do you find the product of two monomials?
Ans: The constant-coefficient of a monomial is multiplied with a constant coefficient of another monomial, and the variable is multiplied with a variable.

Q.2. What are the rules in multiplying monomials?
Ans: We will multiply the coefficients. Next, we will multiply the variables using the rule of the exponent wherever it is required.

Q.3. What is a monomial example?
Ans: It is an algebraic expression that contains only one term.
Example: \(2xy\) is a monomial.

Q.4. What are constant monomials?
Ans: It is a monomial with a constant term, or we can represent it by any number. It can be a negative number, a positive number and can be a fraction. It does not consist of variables. The constant monomials are \(2,\,5,\, – 7,\,\frac{5}{6}\), etc. are the constant monomials.

Q.5. What do you understand about the degree of a monomial?
Ans: The degree of a monomial means the sum of the exponents of the variable/variables present in the expression.

We hope this detailed article on the multiplication of a monomial with a monomial helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!