A monomial is a single term that can include any combination of numbers, variables, exponents, and multiplication. To multiply monomials, we need to be familiar with the laws of exponents.

Here, we will look at a brief summary of the multiplication of monomials. Also, we will explore several examples with answers that will allow us to carefully study the process used to solve monomial multiplication exercises.

## Summary of multiplication of monomials

To multiply monomials, we need to understand and know how to apply the laws of exponents. Let’s see a review of the laws of exponents used in the multiplication of monomials:

### Multiply powers with the same base

To solve this, we have to keep the same base and add the exponents. For example:

$latex ({{a}^4})({{a}^3})={{a}^7}$

### Power of a power

To solve this, we keep the same base and multiply the exponents. For example:

$latex {{({{a}^2})}^3}={{a}^6}$

### Power of a product

We solve the power of a product by finding the power of each factor separately. For example:

$latex {{(3a)}^2}=9{{a}^2}$

Therefore, to multiply monomials, we follow these two steps:

**Step 1:** We multiply the coefficients (numbers).

** Step 1:** We multiply the variables using the laws of exponents if necessary.

## Multiplication of monomials – Examples with answers

Each of the following monomial multiplication examples has its respective solution, which details the reasoning and process used. It is advisable to try to solve the examples before looking at the answer.

**EXAMPLE 1**

Multiply the monomials $latex (4x)(3{{x}^2})$.

##### Solution

**Step 1:** We identify and multiply the coefficients: the coefficients are 4 and 3. Therefore, we have:

$latex 4\times 3=12$

**Step 2: **We multiply the variables: the variables are $latex x$ and $latex {{x}^2}$. When we multiply two terms with the same base, we add the exponents:

$latex (x)({{x}^2})={{x}^3}$

Now, we combine the constants and the variables:

⇒ $latex 12{{x}^3}$

**EXAMPLE 2**

Solve the multiplication: $latex (5x)(4{{x}^3}y)$.

##### Solution

**Step 1:** We multiply the coefficients: the coefficients are 5 and 4. Therefore, we have:

$latex 5\times 4=20$

**Step 2: **We multiply the variables: the variables are $latex x$ and $latex {{x}^3}$. We use the law of the product of exponents to multiply the variables:

$latex (x)({{x}^3})={{x}^4}$

Now, we combine the constants and the variables:

⇒ $latex 20{{x}^4}y$

In this case, we do not multiply the variable *y*, so it remains the same.

**EXAMPLE 3**

Multiply the monomials $latex (5{{x}^3}y)(2{{x}^2}{{y}^5})$.

##### Solution

**Step 1:** We start by multiplying the coefficients: the coefficients are 5 and 2. Therefore, we have:

$latex 5\times 2=10$

**Step 2: **Now, we multiply the variables. We have to multiply the variables with the base *x* and the variables with the base *y* separately. Thus, we have:

$latex ({{x}^3})({{x}^2})={{x}^5}$

$latex (y)({{y}^5})={{y}^6}$

Now, we combine the constants and the variables:

⇒ $latex 10{{x}^5}{{y}^6}$

**EXAMPLE 4**

Multiply the monomials $latex (10{{x}^3}{{y}^2}{{z}^5})(-3{{x}^2}{{y}^3}z)$.

##### Solution

**Step 1:** We have to identify and multiply the coefficients: the coefficients are 10 and -3. Therefore, we have:

$latex 10\times -3=-30$

**Step 2: **We multiply all similar variables separately. We use the law of the product of exponents to obtain the result with each variable:

$latex ({{x}^3})({{x}^2})={{x}^5}$

$latex ({{y}^2})({{y}^3})={{y}^5}$

$latex ({{z}^5})(z)={{z}^6}$

Combining the constants and the variables, we have:

⇒ $latex -30{{x}^5}{{y}^5}{{z}^6}$

**EXAMPLE 5**

Multiply the monomials $latex (4{{x}^3}{{y}^2}z)(2{{x}^2}y{{z}^3})(3{{x}^2}{{y}^2}{{z}^2})$.

##### Solution

**Step 1:** We start by multiplying the coefficients: the coefficients are 4, 2, and 3. Therefore, we have:

$latex 4\times 2\times 3=24$

**Step 2: **We have the variables *x*, *y* and *z*. Thus, we multiply each of the variables separately:

$latex ({{x}^3})({{x}^2})({{x}^2})={{x}^7}$

$latex ({{y}^2})(y)({{y}^2})={{y}^5}$

$latex (z)({{z}^3})({{z}^2})={{z}^6}$

Combining all of this, we have:

⇒ $latex 24{{x}^7}{{y}^5}{{z}^6}$

**EXAMPLE 6**

Solve the multiplication: $latex (-3{{a}^2}{{b}^5}{{c}^3})(4{{a}^3}b{{c}^3})(-5{{a}^4}{{b}^3}{{c}^5})$.

##### Solution

**Step 1:** We have to start with the coefficients: the coefficients are -3, 4, and -5. Therefore, we have:

$latex -3\times 4\times -5=60$

**Step 2: **We multiply the variables separately and use the law of the product of exponents:

$latex ({{a}^2})({{a}^3})({{a}^4})={{a}^9}$

$latex ({{b}^5})(b)({{b}^3})={{b}^9}$

$latex ({{c}^3})({{c}^3})({{c}^5})={{c}^{11}}$

Combining all of this, we have:

⇒ $latex 60{{a}^9}{{b}^9}{{c}^{11}}$

**EXAMPLE 7**

Solve the following operation $latex {{(3{{x}^2}y)}^2}{{(4{{x}^2}{{y}^3})}^3}$.

##### Solution

In this case, we have to start by applying the law of the power of a product and the power of a power of exponents to eliminate the outer exponents:

$latex {{(3{{x}^2}y)}^2}{{(4{{x}^2}{{y}^3})}^3}$

⇒ $latex (9{{x}^4}{{y}^2})(64{{x}^6}{{y}^9})$

**Step 1:** Now, we have to multiply the coefficients: the coefficients are 9 and 64. Therefore, we have:

$latex 9\times 64=576$

**Paso 2: **We have the variables *x* and* y*, so we multiply them separately:

$latex ({{x}^4})({{x}^6})={{x}^{10}}$

$latex ({{y}^2})({{y}^9})={{y}^{11}}$

Now, we combine the constants and the variables:

⇒ $latex 576{{x}^{10}}{{y}^{11}}$

## Multiplication of monomials – Practice problems

Put your knowledge about the multiplication of monomials into practice with the following problems. Solve the problems and select your answer. Check the chosen answer to verify that it is correct. If you need help, you can look at the solved examples above.

## See also

Interested in learning more about operations with polynomials? Take a look at these pages:

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