# Simplifying Exponential Expressions

It's easy to simplify exponential expressions, once you have the basics. What you need to do is to "cancel out" and "group" as many common factors as possible. By common factors, I mean the ones in the number and in the variables. Confused? Just follow the examples and you should get what I mean.

First, go back and review a few basics that you will need from exponentiation and indices that we will use in the examples to follow.

The key thing to remember is that xn (n is called the index) means x multiplied by itself by n times. For example, x3 = x . x . x (by the way, "." is the same thing as multiply, or "×"). If n is a negative number, i.e. x-3, then simplifying it means that x-3 = 1/(x3), or 1/(x . x . x). No cancellation of common factors is done in this case.

Example 1
Simplify:

Example 1 Solution
a) In this case, there's nothing to cancel out. So just group all the multiplied "x's" together.

b) Simplifying a negative indexed exponent means that you need to put it in a form without a negative exponent.

c) As with problem b), you need to put this into a form without a negative exponent.

Example 2
Simplify the following expressions.

Example 2 Solution
There's no grouping to be done here, just cancelling out the common factors.

Example 3
Simplify the following expressions.

Example 3 Solution
(a) In this case, we'll need to cancel out some of the "y's". In addition, "6" and "4" have the common factor "2".

(b) Cancelling out needs to be done for the coefficients (the numbers), and the variables x, y and z.

Example 4
Simplify the following expressions.
(a) (25x2y3)0
(b) -(25x2y3)0

Example 4 Solution
(a) Anything by the power of 0 = 1.
(25x2y3)0 = 1

(b) Again, any number by the power of 0 = 1.
-(25x2y3)0
= -1

Example 5
Simplify the following expressions.

Example 5 Solution

Example 6
Simplify the following expressions

Example 6 Solution
Order isn't important in multiplication, so you can approach this problem in many different ways and still arrive at the same answer. I'll show you 2 different approaches to simplify each exponential expression.

(a) Square the outside, then simplify the result.

OR

Simplify the inside, then square the result.

(b) Multiply, then simplify the result.

OR

Simplify the insides, then multiply the results.

Example 7
Simplify the following

Example 7 Solution
There are many ways to simplify these exponential expressions. I'll illustrate 2 possible solutions for each. Take note that these aren't necessarily the most elegant or efficient solutions. If you can write better ones, please do take the time to send them in. Either type them out, or take a photo of your solution and send it in.

a)
Solution 1:

Solution 2:

b)
Solution 1:

Solution 2:

c)
Solution 1:

Solution 2:

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