Rules of Exponents

 

In this section we're going to review the rules of exponents. This is important because the next part will be over nth roots and other radicals.

Remembering from Algebra 1:

x * x * x * x

is the same as

x⁴

This also applies to coefficients:

10 * 10 * 10 * 10 = 10⁴

This is the most basic rule, but it's also important to know. Now, let's move on to the Addition rule of exponents.

Addition Rule:

If two expressions have the same base, then their exponents can be added.

Example:

3² * 3³ = 3⁵ = 243

You could, if you wanted to check yourself, follow order of operations and do the 8⁸ first, then do 8⁴ and finally add them. The rule just makes it a little easier and faster.

If the two expressions do not have the same base, then they cannot be evaluated.

5² * 3⁵

This cannot be evaluated using the addition rule, it would have to evaluated separately. This rule also applies to variables:

z⁸ * z¹² = z²⁰

 

Roots and Radicals

The most commonly known root is the square root (on here, where ever you see sqrt( ), this means the square root of what ever is in the parentheses). So, we'll look at square roots first followed by nth-roots and then fractional exponents.

When dealing with numbers, there are some that are called 'perfect squares'. These numbers evaluate easily when we find the square root of them. Some perfect squares are: 1, 4, 9, 16, 25, 36, 49, and 64. In order to fully understand roots, we have to be aware of what it means to be the 'root' of something, and since the square root is the easiest I'll use it to illustrate.

When we find the square root of a number, we are simply trying to find two numbers, that when multiplied by itself twice will yield the number you start with; so for example:

64

When you find the square root of the number 64, you are looking for one number, that when multiplied by itself twice will give you 64, thus:

sqrt(64) = 8 because 8 * 8 =64

As a side bar, I may come back later and, after searching around for it, show the actual algorithm that is used in finding the square root for the curious.

This logic can be applied to any root. Before I can explain this further, we need to know some terminology behind roots, take a look at the figure below:

I'll explain this figure in paragraph form as well: the n is what's called the index. This decides to what root you will be taking a number, if there is no index supplied, it is understood to be a 2 which means the square root. The x is some number that you are working with. In our previous example this would have been 64. The white bit that the arrow is pointing to is called the radical, this just denotes the root operation, much like the + denotes addition and - denotes subtraction.

If the index is a 3, then this is sometimes called the cube root of a number.

Now that we have the basics out of the way, let's take a look at fractional exponents. We'll look at a number to the 1/2 power first.

If we have this: 64^1/2this is the same as saying "the square root of 64". So, drawing from this, anytime you see a number raised to the one-half power, it means to simply take the square root of that number. Let's look at how we know this.

This image probably looks a little complex, but it really isn't once broken down. All this is saying is, that the denomiator is the index of the radical and the numerator is what the result is raised to. In the 64^1/2example, it would be sqrt(64)¹ which evaluates to 8¹ which is 8.

!Note: For nth-roots, I'll be using this convention: ^indexroot( ). So, if we had the 4th root of 16 it would look like this: ⁴root(16).

Let's look at two examples.

Example 1:

Evaluate 8^2/3.

First, we write it out using the last formula in the above figure: ³root(8)²

What this is asking is this: What number can be multiplied by itself three times and will equal eight? That's what the first part of the expression is asking. Since I chose a simple base, we can do this one in our heads, but had it been larger a calculator could be used.

In this case, the cube root of 8 is 2 because: 2 * 2 * 2 = 8

Once we have solved the first part of this expression, we then raise the result to the second power: 2² which is 4.

So our answer is 4.

Let's look at another one.

Example 2:

Evaluate 15^5/3.

First we'll write it out like we did above: ³root(15)⁵

We'll evaluate the cube root of 15 first (good time for a calculator) which is: 2.46621207433

We then raise this to the 5th power: 2.46621207433⁵ which is: ~91.2330