Read: Negative and Zero Exponent Rules
Learning Objectives
- Define and use the zero exponent rule
- Define and use the negative exponent rule
Define and use the zero exponent rule
Return to the quotient rule. We worked with expressions for which so that the difference would never be zero or negative.The Quotient (Division) Rule for Exponents
For any non-zero number x and any integers a and b:If we were to simplify the original expression using the quotient rule, we would have
If we equate the two answers, the result is . This is true for any nonzero real number, or any variable representing a real number.
The sole exception is the expression . This appears later in more advanced courses, but for now, we will consider the value to be undefined.
The Zero Exponent Rule of Exponents
For any nonzero real number , the zero exponent rule of exponents states thatExample
Simplify each expression using the zero exponent rule of exponents.Answer: Use the zero exponent and other rules to simplify each expression.
- \Large\begin{array}\text{ }\frac{c^{3}}{c^{3}} \hfill& =c^{3-3} \\ \hfill& =c^{0} \\ \hfill& =1\end{array}
Define and use the negative exponent rule
Another useful result occurs if we relax the condition that in the quotient rule even further. For example, can we simplify ? When —that is, where the difference is negative—we can use the negative rule of exponents to simplify the expression to its reciprocal. Divide one exponential expression by another with a larger exponent. Use our example, .
If we were to simplify the original expression using the quotient rule, we would have
Putting the answers together, we have . This is true for any nonzero real number, or any variable representing a nonzero real number.
A factor with a negative exponent becomes the same factor with a positive exponent if it is moved across the fraction bar—from numerator to denominator or vice versa.
We have shown that the exponential expression is defined when is a natural number, , or the negative of a natural number. That means that is defined for any integer . Also, the product and quotient rules and all of the rules we will look at soon hold for any integer .
The Negative Rule of Exponents
For any nonzero real number and natural number , the negative rule of exponents states thatExample
Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.Answer:
Solution
Combine exponent rules to simplify expressions
In the next examples we will combine the use of the product and quotient rules to simplify expressions whose terms may have negative or zero exponents.Example
Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.Answer: