2.08 Multiplication properties with negative exponents
Ideas
The negative power property
The negative power property states: a^{-x}=\dfrac{1}{a^{x}}
So if you need to express a negative exponent as a positive exponent, or a positive exponent as a negative exponent, you need to convert it to a fraction using this rule.
Examples
Example 1
Express 3^{-3} with a positive exponent.
The negative power property states: a^{-x}=\dfrac{1}{a^{x}}
Raise a fraction to a negative power
Things become a bit more complicated when we need to raise a fraction to a negative power.
Let's first review what a reciprocal is.
The reciprocal of \dfrac{1}{8} is \dfrac{8}{1}=8. The reciprocal of \dfrac{2}{3} is \dfrac{3}{2}. The reciprocal of 91 is \dfrac{1}{91}.
So, if you have a fraction you need to invert it (or flip it) to find the reciprocal. If you have an integer, you put that integer as the denominator, and 1 as the numerator.
Consider \left(\dfrac{a}{b} \right)^{-1}. We can write this without negative exponents, by applying the negative power to the terms inside the parentheses and applying the negative power property.
| \displaystyle \left(\dfrac{a}{b} \right)^{-1} | \displaystyle = | \displaystyle \dfrac{a^{-1}}{b^{-1}} | Apply the power to the terms inside the parentheses |
| \displaystyle = | \displaystyle a^{-1} \div b^{-1} | Rewrite as division | |
| \displaystyle = | \displaystyle \dfrac{1}{a} \div \dfrac{1}{b} | Apply the negative exponent | |
| \displaystyle = | \displaystyle \dfrac{1}{a} \times \dfrac{b}{1} | Multiply by the reciprocal | |
| \displaystyle = | \displaystyle \dfrac{b}{a} | Simplify |
In the question below we will see that: \left(\dfrac{a}{b} \right)^{-n}=\left(\dfrac{b}{a}\right)^n=\dfrac{b^n}{a^n}.
Examples
Example 2
Write \left(\dfrac{2}{5}\right)^{-2} with positive exponents.
Example 3
Express \left(\dfrac{3^{2}}{4^{4}}\right)^{-1} using positive exponents.
When putting a fraction to a negative exponent we get:
\left(\dfrac{a}{b} \right)^{-1}=\dfrac{b}{a} and \left(\dfrac{a}{b} \right)^{-n}=\left(\dfrac{b}{a}\right)^n=\dfrac{b^n}{a^n}.
Multiplication property with negative exponents
Now we are going to combine these rules to simplify expressions which involve multiplication properties and negative exponents.
Consider the expression: 2^{7} \times 2^{-4}
Notice the following:
There is multiplication and the bases are the same (we can apply the product of powers property )
One of the powers is negative (we can express the second term with a positive power if we wish)
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 choices:
One approach: Add the powers immediately as the bases are the same and we are multiplying.
| \displaystyle 2^{7} \times2^{-4} | \displaystyle = | \displaystyle 2^{7+-4} |
| \displaystyle = | \displaystyle 2^{7-4} | |
| \displaystyle = | \displaystyle 2^3 |
Another approach: First express the second term with a positive power.
| \displaystyle 2^{7} \times 2^{-4} | \displaystyle = | \displaystyle 2^{7} \times \dfrac{1}{2^{4}} | |
| \displaystyle = | \displaystyle \dfrac{2^{7}}{2^{4}} | ||
| \displaystyle = | \displaystyle 2^{7-4} | (subtract the powers using the division property of exponents) | |
| \displaystyle = | \displaystyle 2^{3} |
Choose the approach that makes the most sense to you.
Examples
Example 4
Write the following with negative exponents: \dfrac{1}{6} \times \dfrac{1}{6} \times \dfrac{1}{6} \times \dfrac{1}{6}
Example 5
Express the following with positive exponents. 5^{2}\times2^{-4}\times 2^{-2} \times5^{6}
Multiplication properties with negative exponents work 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.