## 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.