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

1) \(-3x^2y^3z \ \times \ 3x = \)\( \ \color{red}{-9x^3y^3z}\)

2) \(4x^2y^3z \ \times \ 3x = \)\( \ \color{red}{12x^3y^3z}\)

3) \(1xy \ \times \ 2x^2y = \)\( \ \color{red}{2x^3y^2}\)

4) \(6x^2y^2z \ \times \ 3xz^2 = \)\( \ \color{red}{18x^3y^2z^3}\)

5) \(8xy \ \times \ 2x^2y = \)\( \ \color{red}{16x^3y^2}\)

6) \(-1x^2y^2z \ \times \ 4xz^2 = \)\( \ \color{red}{-4x^3y^2z^3}\)

7) \(3xy \ \times \ (-3z) = \)\( \ \color{red}{-9xyz}\)

8) \(10xy \ \times \ (-2z) = \)\( \ \color{red}{-20xyz}\)

9) \(-4xy \ \times \ (-2z) = \)\( \ \color{red}{8xyz}\)

10) \(-8x^2y^2z \ \times \ 7xz^2 = \)\( \ \color{red}{-56x^3y^2z^3}\)

How to Multiply Monomials