How to Multiply Monomials

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The phrase "monomial," as the name implies, is made up of the terms "mono" and "nomial," which relate to a single expression. A monomial is an algebraic or mathematical phrase consisting of just one term with no mathematical operation. Furthermore, the monomial terms can be variables or a mixed term with a co-efficient. Furthermore, the variable's power (let's say $x$ ) in a monomial should be an integer, not a fractional power like the squared or cube root.
For example, $2x^{\frac{1}{3}}$ is not a monomial as the term has a fraction power of $x$ . Now, monomial is a type of polynomial, about which we will learn next.

Degree of a Monomial

Unlike a polynomial, the degree of a monomial is pretty simple to find. Since a monomial consists of only one term, hence its degree would obviously be the exponent power of that term. So, for example in the term $4x^3$ , we can see that the degree of $x$ is $3$ . So, the degree of the monomial is considered as $3$ .
Now, for a monomial expression with multiple variables, the case becomes quite complex. So, for example, let’s take the multivariable monomial expression $3x^2y^3$ . Now, here we can clearly see that the monomial has 2 different variables as $x$ and $y$ . So, what we do here is add up the exponent powers of the separate variables. So, in this case the addition comes as $2 \ + \ 3 \ = \ 5$ . Hence, the degree of this monomial is $5$ .

Some Rules in Multiplying Monomials

While multiplying monomials, keep the following rules in mind:

Always remember the sign rules while multiplication. If both signs are the same, the resulting sign would be a “+”. And, if both signs are opposite, then the resulting sign would be “-“.
Next, while multiplying, the powers of the same variables add up. For example, the multiplication of $3x^2 \times 2x^3$ gives $6x^5$ as a result. So, here we can see that the powers $2$ and $3$ added up to give $5$ .

Multiplying Monomials

Let’s understand the multiplication in monomials by the following examples.

$3x^2y \times -3x \ = \ 3x^2y \times -3x \ = \ -9x^3y$
$3x^2 \times 12x \ = \ 3x^2 \times 12x \ = \ 36x^3$
$y \times -9x \ = \ y \ -9x \ = \ -9xy$
$4x^2 \times -3x \ + \ 7x^3 \times 6x^2 \ – \ 5x \times -6x \ = \ -12x^3 \ + \ 42x^5 \ + \ 30x^2$

Free printable Worksheets

Exercises for Multiplying Monomials

1) $-3x^2y^3z \ \times \ 3x =$

2) $4x^2y^3z \ \times \ 3x =$

3) $1xy \ \times \ 2x^2y =$

4) $6x^2y^2z \ \times \ 3xz^2 =$

5) $8xy \ \times \ 2x^2y =$

6) $-1x^2y^2z \ \times \ 4xz^2 =$

7) $3xy \ \times \ (-3z) =$

8) $10xy \ \times \ (-2z) =$

9) $-4xy \ \times \ (-2z) =$

10) $-8x^2y^2z \ \times \ 7xz^2 =$

Answers

- 3 x^{2} y^{3} z \times 3 x =

$-3x^2y^3z \ \times \ 3x =$

- 9 x^{3} y^{3} z

$\ \color{red}{-9x^3y^3z}$

4 x^{2} y^{3} z \times 3 x =

$4x^2y^3z \ \times \ 3x =$

12 x^{3} y^{3} z

$\ \color{red}{12x^3y^3z}$

1 x y \times 2 x^{2} y =

$1xy \ \times \ 2x^2y =$

2 x^{3} y^{2}

$\ \color{red}{2x^3y^2}$

6 x^{2} y^{2} z \times 3 x z^{2} =

$6x^2y^2z \ \times \ 3xz^2 =$

18 x^{3} y^{2} z^{3}

$\ \color{red}{18x^3y^2z^3}$

8 x y \times 2 x^{2} y =

$8xy \ \times \ 2x^2y =$

16 x^{3} y^{2}

$\ \color{red}{16x^3y^2}$

- 1 x^{2} y^{2} z \times 4 x z^{2} =

$-1x^2y^2z \ \times \ 4xz^2 =$

- 4 x^{3} y^{2} z^{3}

$\ \color{red}{-4x^3y^2z^3}$

$3xy \ \times \ (-3z) =$

$\ \color{red}{-9xyz}$

$10xy \ \times \ (-2z) =$

$\ \color{red}{-20xyz}$

$-4xy \ \times \ (-2z) =$

$\ \color{red}{8xyz}$

10)

$-8x^2y^2z \ \times \ 7xz^2 =$

$\ \color{red}{-56x^3y^2z^3}$

How to Multiply Monomials