Multiplying and Dividing Monomials

 

Learning Objective(s)

·         Multiply and divide monomials.

 

 

A monomial is an expression that consists of a number, a variable, or the product of numbers and variables. Expressions such as 2, z, and 42p³y 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 coefficient.  For example, in the monomial 8x², 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.

 

 

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.

 

 

 

Area of square = (2x)(2x) =

 

The area, 4x², is a product of a number (4) and a variable with a whole number exponent (x²). 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 = pr², 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 p.

 

 

 

So the area of the circle with radius 2xy is 4px²y². This is a monomial with a coefficient of.

 

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:

 

 

 

 

To simplify a power of a power, we multiply the exponents. For example, (2³)⁵ = 2¹⁵. That same rule applies to variables:

 

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.

 

 

 

 

 

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:

 

 

 

Multiply the monomials and express the answer in simplest terms.     A)   B)      C)   D)   Show/Hide Answer A) Incorrect. The first monomial has been raised to a power, but the two monomials haven't been multiplied together.. The correct answer is .   B) Incorrect. The fractional coefficient has not been simplified. The correct answer is .   C) Correct. Each variable appears only once, fractions are written in lowest terms, coefficients have been multiplied, and there are no powers of powers.   D) Incorrect. Incorrect.  The exponent for the base b is incorrect.  When you take a power of a power, you multiply exponents turning  into, then when multiplying factors with the same base you add the exponents, turning  into . The correct answer is .

 

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

 

 

 

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

 

 

 

 

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 or 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.