Section 5.2 learning objectives
5.2: Exponent Properties Used Together
- Simplify expressions using a combination of exponent rules
- Simplify expressions with negative exponents using a combination of exponent rules
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. 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 4 for x.
Example 1
Evaluate [latex] \displaystyle \frac{24{{x}^{8}}}{2{{x}^{5}}}[/latex] when [latex]x=4[/latex].
Example 2
Evaluate [latex] \displaystyle \frac{24{{x}^{8}}{{y}^{2}}}{{{(2{{x}^{3}}y)}^{2}}}[/latex] when [latex]x=4[/latex] and [latex]y=-2[/latex].
Notice that you could have worked this problem by substituting 4 for x and -2 for y in the original expression. You would still get the answer of 96, but the computation would be much more complex. Notice that you didn’t even need to use the value of y 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.
Example 3
Simplify. [latex]a^{2}\left(a^{5}\right)^{3}[/latex]
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.
Example 4
Simplify. [latex] \displaystyle \frac{{{a}^{2}}{{({{a}^{5}})}^{3}}}{8{{a}^{8}}}[/latex]
Example 5
Simplify: [latex] (3a^5b^2c^2)^2 (2a^4b^3c)^3[/latex]
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.
Rules for Negative Exponents
With a, b, m, and n not equal to zero, and m and n as integers, the following rules apply:
[latex]\displaystyle a^{-m}=\frac{1}{a^{m}}[/latex]
[latex]\displaystyle\frac{1}{a^{-m}}=a^{m}[/latex]
[latex]\displaystyle\frac{a^{-n}}{b^{-m}}=\frac{b^m}{a^n}[/latex]
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.
| Rewrite with positive Exponents First | Description of Steps Taken | Apply the Product Rule for Exponents First | Description of Steps Taken |
| [latex] \frac{\left(4x^{3}\right)^{5}}{\left(2x^{2}\right)^{4}}[/latex] | move the term [latex]{{\left( 2{{x}^{2}} \right)}^{-4}}[/latex] to the denominator with a positive exponent | [latex] \left(4^5x^{15}\right)\left(2^{-4}x^{-8}\right)[/latex] | Apply the exponent of 5 to each term in expression on the left, and the exponent of -4 to each term in the expression on the right. |
| [latex]\frac{\left(4^5x^{15}\right)}{\left(2^4x^{8}\right)}[/latex] | Use the product rule to apply the outer exponents to the terms inside each set of parentheses. | [latex]\left(4^5\right)\left(2^{-4}\right)\left(x^{15}\cdot{x^{-8}}\right)[/latex] | Regroup the numerical terms and the variables to make combining like terms easier |
| [latex]\left(\frac{4^5}{2^4}\right)\left(\frac{x^{15}}{x^{8}}\right)[/latex] | Regroup the numerical terms and the variables to make combining like terms easier | [latex]\left(4^5\right)\left(2^{-4}\right)\left(x^{15-8}\right)[/latex] | Use the rule for multiplying terms with exponents to simplify the x terms |
| [latex]\left(\frac{4^5}{2^4}\right)\left(x^{15-8}\right)[/latex] | Use the quotient rule to simplify the x terms | [latex]\left(\frac{4^5}{2^4}\right)\left(x^{7}\right)[/latex] | Rewrite all the negative exponents with positive exponents |
| [latex]\left(\frac{1,024}{16}\right)\left(x^{7}\right)[/latex] | Expand the numerical terms | [latex]\left(\frac{1,024}{16}\right)\left(x^{7}\right)[/latex] | Expand the numerical terms |
| [latex]64x^{7}[/latex] | Divide the numerical terms | [latex]64x^{7}[/latex] | Divide the numerical terms |
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.
Example 6
Simplify [latex]\displaystyle\frac{\left(t^{3}\right)^2}{\left(t^2\right)^{-8}}[/latex]
Write your answer with positive exponents.
Example 7
Simplify [latex]\displaystyle\frac{\left(5x\right)^{-2}y}{x^3y^{-1}}[/latex]
Write your answer with positive exponents.
The video below works through a few more examples of simplifying expressions that contain negative exponents.
These last examples are fairly complex. We will use multiple exponent properties and follow the sequence of steps below to simplify each of these expressions. However, as was shown above, you don’t always need to follow the same sequence to get the correct result.
- Clear any parentheses by using the Power Rule.
- Apply the definition of a negative exponent and rewrite negative exponents as positive exponents.
- Simplify the numerator. (Combine like bases by using the Product Rule. Simplify numerical bases.)
- Simplify the denominator. (Combine like bases by using the Product Rule. Simplify numerical bases.)
- Reduce, if possible. (Reduce coefficients. Combine like bases using the Quotient Rule.)
Example 8
Simplify: [latex]\displaystyle\frac{6a^4b^{-5}}{6^{-1}a^{-1}b^4}[/latex]
Example 9
Simplify: [latex]\displaystyle{\left(\frac{-3x^{-2}yz^{-1}}{6x^{-1}y}\right)}^2[/latex]
The following video works through one more example of simplifying an expression containing negative exponents.