 ## How to Multiply Monomials

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.

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

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

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

## Multiplying Monomials Practice Quiz

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