The Rules of Exponents


Before you can get into the nitty-gritty details of polynomials, you’ve got to start somewhere simpler. Here we’re going to take a look at some basic rules of exponents that you need to know.

First, what if we multiply two values together that have exponents? Like what if we try to solve

x^4 * x^7 

What we have to remember is that x^4 is just x * x * x * x, and x^7 is just x * x * x * x * x * x * x. With this in mind, we can do a little simplifying and figure out a general rule:

(x * x * x * x) * (x * x * x * x * x * x * x) 

From here, we can drop the parentheses because order doesn’t matter for multiplication because of the commutative property. That gives us

x * x * x * x * x * x * x * x * x * x * x 

Which, by definition, is

x^11 

So a general rule we can get from this is:

x^a * x^b = x^{a+b} 

So for a few examples:

x^3 * x^2 = x^{3+2} = x^5 
5^3 * 5^7 = 5^{3+7} = 5^{10} 
w^4 * w^x = w^{4+x} 

If we have division, a similar rule applies:

\frac{x^a}{x^b} = x^{a-b} 
So again, for a few examples:

\frac{x^{12}}{x^5} = x^{12-5} = x^7 
\frac{3^{15}}{3^2} = 3^{13} 
\frac{y^7}{y^a} = x^{7-a} 

There is one last rule to learn, and that’s if you have a term with an exponent taken to another exponent. Here is the pattern:

(x^a)^b = x^{ab} 

Here follows a few examples of this pattern in action:

(x^2)^4 = x^{2*4} = x^8 
(y^3)^x = y^{3x} 
(2^7)^w = 2^{7w} 

Keeping these general rules for exponents in mind will help you to work through the rest of Algebra 1 without many hangups. Learning them well means that you will breeze through all polynomial situations where you are dealing with exponents, and this will make life much easier on you.