# More on Exponents

So now we know that ${x}^{3}$ is shorthand math notation for x × x × x. In exponents, the x is called the base and the 3 is called the exponent or the power of x. Anything to the power of 2 is referred to as squared and anything to the power of 3 is referred to as cubed. These terms come from the formulas for finding the areas of a square and the volume of a cube.

There are several rules involving exponents that allow us to simplify expressions. So let’s look at

$\begin{array}{c} {{x}^{2}\hspace{0.33em}\times\hspace{0.33em}{x}^{3}}\\ {\Longrightarrow\hspace{0.33em}{x}^{2}\hspace{0.33em}\times\hspace{0.33em}{x}^{3}\hspace{0.33em}{=}\hspace{0.33em}{(}{x}\hspace{0.33em}\times\hspace{0.33em}{x}{)(}{x}\hspace{0.33em}\times\hspace{0.33em}{x}\hspace{0.33em}\times\hspace{0.33em}{x}{)}\hspace{0.33em}{=}\hspace{0.33em}{x}^{5}} \end{array}$

So it appears that you can get the final result simply by adding the exponents together, this is correct as long as the exponents apply to the same base. For any two numbers a and b:

${x}^{a}\hspace{0.33em}\times\hspace{0.33em}{x}^{b}\hspace{0.33em}{=}\hspace{0.33em}{x}^{a}{x}^{b}\hspace{0.33em}{=}\hspace{0.33em}{x}^{{a}{+}{b}}$

Now you remember that when you see x, there is an implied “1” in front of it. Well, when dealing with exponents, if there is no visible exponent, there is an implied “1” there as well since ${x}^{1}\hspace{0.33em}{=}\hspace{0.33em}{x}$. So what about a rule for division?

$\frac{{x}^{3}}{{x}^{2}}\hspace{0.33em}{=}\hspace{0.33em}\frac{\rlap{/}{x}\hspace{0.33em}\times\hspace{0.33em}\rlap{/}{x}\hspace{0.33em}\times\hspace{0.33em}{x}}{\rlap{/}{x}\hspace{0.33em}\times\hspace{0.33em}\rlap{/}{x}}\hspace{0.33em}{=}\hspace{0.33em}\frac{x}{1}\hspace{0.33em}{=}\hspace{0.33em}{x}^{1}\hspace{0.33em}{=}\hspace{0.33em}{x}$

Because of the common factors of x in the numerator and denominator, I can cancel these away, leaving “1” in their place. So it appears that to get the final result, you just subtract the exponent in the denominator from the one in the numerator. Again, this is correct as long as the base is the same:

$\frac{{x}^{a}}{{x}^{b}}\hspace{0.33em}{=}\hspace{0.33em}{x}^{{a}{-}{b}}$

There are a few more things about exponents I want to cover before we use these in equations. This will be covered in my next post.