When you multiply one monomial by another, the result is also a monomial. A
monomial
is a mathematical expression that consists of one term only. Examples of monomials include x, y, 3x, 4y, x
2
, y
2
, and so on. Note that a monomial cannot have negative exponents.
If you multiply one monomial by another, you will get a monomial as the result. The coefficients (the numbers in front of the variables) of the monomials are multiplied together, and then the variables are multiplied. For instance, if you multiply the monomials 3x and 4y, the product is 12xy. If both monomials have the same variables with the same exponents, then the
laws of exponents
apply.
Remember: The result of multiplying monomials is always a monomial.
How to Multiply Two Monomials
As mentioned above, when two monomials are multiplied, the result is a monomial. Let's look at some examples to illustrate this concept:
Examples:
1. 4x × 5y
Break down the terms.
⇒ 4 × x × 5 × y
Multiply the coefficients and variables separately, we get;
⇒ (4 × 5) × (x × y)
⇒ 20 × xy
⇒ 20xy
2. 6x × 7y
Break down the terms.
⇒ 6 × x × 7 × y
Multiply the coefficients and variables separately, we get;
⇒ (6 × 7) × (x × y)
⇒ 42 × xy
⇒ 42xy
3. 3x × (–4y)
Break down the terms.
⇒ 3 × x × (–4) × y
Multiply the coefficients and variables separately, we get;
What if we need to multiply more than two monomials? The process is straightforward: first, multiply the first two monomials, and then multiply the result by the next monomial. This process can be repeated for any number of monomials. Let's see some examples.
Examples:
1. 3x × 4y × 5z
First, multiply the monomials 3x and 4y.
⇒ (3x × 4y) × 5z
⇒ 12xy × 5z
Now multiply the product with the third monomial
⇒ 12 × x × y × 5 × z [Separating each term]
⇒ (12 × 5) × x × y × z [Multiplying the coefficients and variables separately]
⇒ 60 xyz
2. xy × 3x
2
y
2
× 4x
3
y
3
First, multiply the monomials xy and 3x
2
y
2
.
= (xy × 3x
2
y
2
) × 4x
3
y
3
Using the laws of exponents, where a
m
× a
n
= a
mn
, we get:
= 3 x
3
y
3
× 4x
3
y
3
Now multiply the product with the third monomial
= 3 × x
3
× y
3
× 4 × x
3
× y
3
[Separating each term]
= 3 × 4 × x
3
× x
3
× y
3
× y
3
[Multiplying the coefficients and variables separately]
= 12 × x
6
× y
6
= 12 x
6
y
6
Sample Problems and Solutions
Q.1: Multiply x and 2x
3
y
Solution: Given two monomials are x and 2x
3
y
The product of two monomials:
⇒ x × 2x
3
y
⇒ x × 2 × x
3
× y [Separating each term]
⇒ 2 × x × x
3
x y [Multiplying the coefficients and variables separately]
⇒ 2 × x
4
× y [By laws of exponents, a
m
× a
n
= a
mn
]
⇒ 2x
4
y
Q.2: Find the volume of a cuboid, where length = 2x, breadth = 3y, and height = 4z.
Solution: The dimensions of the cuboid are:
Length = 2x
Breadth = 3y
Height = 4z
We'll denote the volume as V.
We know that the volume of a cuboid is given by the formula:
Volume = Length × Breadth × Height
V = 2x × 3y × 4z
Multiply the first two monomials,
V = (2x × 3y) × 4z
V = 6xy × 4z
V = 6 × 4 × x × y × z
V = 24 xyz cubic units.
Therefore, the volume of the cuboid is 24 xyz cubic units.
Practice Questions
Multiply 4x × 5x
2
Multiply 6x × (– 3xyz)
Find the volume of a rectangular box with the given length, breadth, and height, respectively.
3a, 5b, 7a
m
2
n × n
2
p × p
2
m
3x × 5x
2
× 7x
3
Multiply the three monomials, 5a
4
t
7
, 4t
3
a
6
k
2
, 7t
2
3k
4
f
5
Find the product of 5xy and 6yz.
Learn more about different types of polynomials and how to perform operations on them.
When two monomials are multiplied, then multiply the coefficients and multiply the variables, separately. Also, use the laws of exponents wherever required.
How to multiply three monomials?
Find the product of the first two monomials and then multiply the product with the third monomial.
What is the result of multiplication of monomials?
The multiplication of monomials results in a monomial only.
What is the result of multiplying a monomial by a binomial? Give an example.
If we multiply a monomial by a binomial, then we need to use the distributive property and the product will be a binomial. For example, 3x multiplied by (2x +y) we get; 3x × (2x + y) = 3x × 2x + 3x × y = 6x^2 + 3xy.
Is 2x a single term or two terms?
2x is a single term, hence a monomial.
Can a monomial have more than one variable?
Yes, a monomial can have more than one variable such as xy, xyz, x^2y^2z^2, etc. are monomials.