Some Exponent Rules & Tips

First, what is an exponent? That is the fancy math term for power or degree, we have special terms to use for 2 & 3, which are squared & cubed, so they may sound familiar. They are also the most common since they also get used in other settings, such as feet squared or cubic yards, etc. And the do come from this exponent idea.

So what does it mean? It just means that we are multiplying something by itself a certain number of times, sometimes this is referred to as repeated multiplication. That thing we are multiplying is called the base, and the number telling us how many times is the exponent. This works with numbers, variables, units of measure, and some other items. In this course we are really only working with them using numbers and variables.

So if we have 43, also 4^3, it just means 4*4*4, which is 64. The ^ is above the 6 on the keyboard if you can’t find the superscript. You always need the number after the ^ to tell you how many times to do that multiplication.

If you have anything after the item to the power, it is a good idea, especially when typing using the ^, to have parentheses to clarify what you mean.

If you type 3^45^6, it looks like you mean 3^(45^6), but you may have meant (3^4)(5^6). This is also true when you have variables in the mix. So keep in mind your Order of Operations!

Now there are also some rules that make working with Exponents easier. I am only going to go over a couple of them here.

Product Rule, same base multiplied together, just add the exponents: $a^m\ast a^n=a^{(m+n)}$
$4^3\ast 4^4=4^{(3+4)}=4^7$
Quotient Rule, same base, but division, subtract the exponents: $\frac{a^m}{a^n}=a^{m-n}$
$\frac{4^7}{4^3}=4^{7-3}=4^4$
Power of a product (or a quotient), basically distribute the power to each thing in the parentheses: $\left(a\ast b\right)^m=a^m\ast b^m$ or $\left(\frac ab\right)^m=\frac{a^m}{b^m}$
$\left(5\ast7\right)^8=5^8\ast7^8$
Power Rule, that is the power of a power, multiply the exponents: $\left(a^m\right)^n=a^{m\ast n}$
$\left(5^4\right)^3=5^{4\ast 3}=5^12$

I used numbers in the above examples since we already have letters in the general rule. You can have variables, expressions, all sorts of things happening. Just follow all of our rules!

~ Dr. Tammy