2.09 Division properties with negative exponents
Introduction
We previously looked at the negative power property , which states: a^{-x}=\dfrac{1}{a^{x}}. We'll now look at how to apply this to fractions and division problems.
Division property with negative exponents
We've already learned about the properties for exponents with division , as well as the negative power property. Now we are going to combine these rules to simplify expressions which involve multiplication or division, and negative exponents.
Consider the expression: 2^{6} \div 2^{-2}
Notice the following:
There is division and the bases are the same (we can apply the division property of exponents)
One of the powers is negative (we can express the second term with a positive power if we want to)
When negative powers are involved, this opens up choices in how we go about trying to simplify the expression.
With the above example, we have two ways that we can solve:
One approach: subtract the powers immediately as the bases are the same and we are dividing.
| \displaystyle 2^{6} \div 2^{-4} | \displaystyle = | \displaystyle 2^{6-(-4)} |
| \displaystyle = | \displaystyle 2^{6+4} | |
| \displaystyle = | \displaystyle 2^{10} |
Another approach: first express the second term with a positive power.
| \displaystyle 2^{6} \div 2^{-4} | \displaystyle = | \displaystyle 2^{6} \div \dfrac{1}{2^{4}} |
| \displaystyle = | \displaystyle 2^{6} \times 2^{4} | |
| \displaystyle = | \displaystyle 2^{6+4} | |
| \displaystyle = | \displaystyle 2^{10} |
Choose the approach that makes the most sense to you.
Examples
Example 1
Express the following with a positive exponent: \dfrac{9^{3}}{9^{4}\times 3^{-4}}
The division property with negative exponents works the same way as with positive exponents, we just have the added step of applying the negative power property at some point in our working.