Simplify expressions using a combination of exponent rules

Once the rules of exponents are understood, you can begin simplifying more complicated expressions. There are many applications and formulas that make use of exponents, and sometimes expressions can get pretty cluttered.  We already looked at how to simplify exponential expressions in the previous section, but we are now going to show some more complex examples of exponential expressions and look at some strategies for how we can successfully use the properties of exponents to simplify these expressions.

Simplifying an expression before evaluating can often make the computation easier, as you will see in the following example which makes use of the quotient rule to simplify before substituting [latex]4[/latex] for [latex]x[/latex].

Notice that you could have worked this problem by substituting [latex]4[/latex] for [latex]x[/latex] and [latex]2[/latex] for [latex]y[/latex] in the original expression. You would still get the answer of [latex]96[/latex], but the computation would be much more complex. Notice that you didn’t even need to use the value of [latex]y[/latex] to evaluate the above expression.

In the following video you are shown examples of evaluating an exponential expression for given numbers.

Usually, it is easier to simplify the expression before substituting any values for your variables, but you will get the same answer either way. In the next examples, you will see how to simplify expressions using different combinations of the rules for exponents.

The following examples require the use of all the exponent rules we have learned so far. Remember that the product, power, and quotient rules apply when your terms have the same base.

Simplify Expressions With Negative Exponents

Now we will add the last layer to our exponent simplifying skills and practice simplifying compound expressions that have negative exponents in them. It is standard convention to write exponents as positive because it is easier for the user to understand the value associated with positive exponents, rather than negative exponents.

Use the following summary of negative exponents to help you simplify expressions with negative exponents.

When you are simplifying expressions that have many layers of exponents, it is often hard to know where to start. It is common to start in one of two ways:

• Rewrite negative exponents as positive exponents
• Apply the product rule to eliminate any “outer” layer exponents such as in the following term: [latex]\left(5y^3\right)^2[/latex]

We will explore this idea with the following example:

Simplify. [latex]\displaystyle {{\left( 4{{x}^{3}} \right)}^{5}}\cdot \,\,{{\left( 2{{x}^{2}} \right)}^{-4}}[/latex]

Write your answer with positive exponents. The table below shows how to simplify the same expression in two different ways, rewriting negative exponents as positive first, and applying the product rule for exponents first. You will see that there is a column for each method that describes the exponent rule or other steps taken to simplify the expression.

If you compare the two columns that describe the steps that were taken to simplify the expression, you will see that they are all nearly the same, except the order is changed slightly. Neither way is better or more correct than the other, it truly is a matter of preference.

In the next section, you will learn how to write very large and very small numbers using exponents. This practice is widely used in science and engineering.