Multiplying and Dividing Monomials

Learning Objective

Introduction

A monomialA number, a variable, or a product of a number and one or more variables with whole number exponents, such as `-5`, `x`, and `8xy^3`. is an expression that consists of a number, a variable, or the product of numbers and variables. Expressions such as `2`, `z`, and `42p^3y` are monomials, while those with more than one term, like `2 + z`, are not.

When monomials include both a number and a variable, the number is called the coefficientA number that multiplies a variable.. For example, in the monomial `8x^2`, `8` is the coefficient.

The variables in a monomial can have whole number exponents, but no negative exponents. Just as numbers can be multiplied and divided, monomials with variables can also be multiplied and divided following the same rules.

Multiplying Monomials

Let's begin by multiplying a simple monomial. Consider a square whose length is `2x`. To find the area of this square, we multiply the side length by itself, or square it.

`2x`The image shows a square with side length 2 x.

Area of square `= (2x)(2x)=2*2*x*x=4x^2`

The area, `4x^2`, is a product of a number (`4`) and a variable with a whole number exponent (`x^2`). In other words, it's a monomial, too. So the result of multiplying two monomials is—another monomial!

Let's try a slightly more complicated problem. Let's find the area of a circle with a radius of `2xy`. The formula for the area of a circle is `A = pir^2`, where `A=` area and `r =` radius. To find the area of a circle with a radius of `2xy`, we'll first need to square the radius, and then multiply by `pi`.

Example

Problem

Find the area of a circle whose radius is `2xy`.

 

`A=pir^2`

 

Write the formula for the area of a circle.

 

`pi*2xy*2xy`

 

Substitute `2xy` for radius.

 

`2*2*x*x*y*y*pi`

 

 

Expand `2xy` and use the commutative property of multiplication.

 

`4x^2y^2pi`

 

Multiply the coefficients and variables.

 

`4pix^2y^2`

 

`pi` is a number, not a variable, so include it with the coefficient.

Answer

`4pix^2y^2`

 

 

           

So the area of the circle with radius `2xy` is `4pix^2y^2`. This is a monomial with a coefficient of `4pi`.

When multiplying monomials, multiply the coefficients together, and then multiply the variables together. If two variables have the same base, add the exponents, like this:

`5a^4*7a^6=35a^10`

Monomials to a Power

To simplify a power of a power, we multiply the exponents. For example, `(2^3)^5=2^15`. That same rule applies to variables: `(x^2)^4=x^8`

When there is a coefficient, or more than one variable, raised to a power of a power, each variable or number is taken to the power.

Example

Problem

Simplify `(x^3y^5)^2`.

 

`(x^3)^2(y^5)^2`

 

Square each variable within the parentheses individually.

 

`x^6y^10`

 

 

Multiply exponents to simplify.

Answer

`x^6y^10`

 

 

 

Example

Problem

Simplify `(-5a^3)^2`.  

 

`(-5)^2(a^3)^2`

 

Square each variable or coefficient within the parentheses individually.

 

`25a^6`

 

Square the coefficient and multiply exponents to simplify.

Answer

`25a^6`

 

 

To simplify the product of two monomials, make sure that each variable appears only once, fractions are written in lowest terms, coefficients have been multiplied, and there are no powers of powers. Below is an example to illustrate these steps:

Example

Problem

Simplify `(1/2xy^2)^3 (-2x^2)`. 

 

`(1/2)^3(x)^3(y^2)^3(-2x^2)`

 

Raise each part of the first monomial to a power of `3`.

 

`1/8x^3y^6(-2x^2)`

 

Simplify the powers.

 

`((-2)/8x^5y^6)`

 

 

Multiply the coefficients and the variables of the two monomials.

 

`-1/4x^5y^6`

 

Rewrite the fraction in lowest terms.

 

Answer

`-1/4x^5y^6`

 

 

 

Multiply the monomials and express the answer in simplest terms.

`(2/3a^2b^3)^2(-1/2ab)`

 

A) `(4/9a^4b^6)(-1/2ab)`

 

B) `-4/18a^5b^7` 

 

C) `-2/9a^5b^7`

 

D) `-2/9a^5b^6` 

 

Dividing Monomials

When dividing monomials, divide the coefficients and then divide variable powers with the same base by subtracting the exponents. Consider this example:

Example

Problem

A rectangle has an area of `8x^2` and a length of `4x`. Find the width of the rectangle.

 

`"area"/"length"="width"`

 

 

Rewrite the area of a rectangle formula to solve for width.

 

`(8x^2)/(4x)`

 

Substitute known values.

 

`2x`

 

 

Divide coefficients, and subtract the exponents of the variables.

Answer

`"width" = 2x`

 

 

That wasn't too hard. Let's try another:

Example

Problem

Simplify `(10x^3y^5)/(2xy^2)`.  

 

`(10/2)(x^3/x)(y^5/y^2)`

 

 

Separate the monomial into numerical and variable factors.

 

`5(x^(3-1))(y^(5-2))`

 

 

Divide the numbers, and subtract the exponents of matching variables.

Answer

`5x^2y^3`

 

 

Summary

Multiplying and dividing monomials are done by following the rules of exponents. To multiply monomials, multiply coefficients and add the exponents of like bases. To raise a monomial to a power, when there is a coefficient of more than one variable raised to a power of a power, each variable or number is taken to the power by multiplying the exponent of the base by the exponent of the power it is being raised to. To divide monomials, divide the coefficients and subtract the exponents of like bases.