Explore the Lesson

B. Negative and Zero Exponent Laws

Simplifying expressions containing negative exponents is a new concept. However, the process is a continuation of the exponent laws you've already worked with in previous math courses. These laws were reviewed in the Warm Up section of this lesson.

Negative Exponent
the index value of a power is a negative number


Using the Quotient of Powers Law, a pattern to recognize how negative exponents are formulated can be examined. Begin with a positive exponent of 3 and approach the negative exponents by decreasing the index value by one every time.

How can a3 be formed using the Quotient of Powers Law of Exponents?

Following the pattern above, a2 and a1 can also be formed using the Quotient of Powers Law.

Similarly, a0 results from the same pattern.

Note the power a0 is related to another exponent law called the Zero Exponent Law. Simplifying fractions that have same value in the numerator and the denominator result in the answer of 1.

Key Lesson Marker

In general, the Zero Exponent Law states any base having an exponent of 0 is equal to 1.

a0 = 1

The pattern created so far is shown below.

How would the pattern work if the next exponent in the pattern is a-1?

Recall that when the value zero is in the denominator of a fraction, the fraction is not defined or does not exist.

Applying the Zero Exponent Law, substitute 1 for a0 and restrict the a so it does not equal zero.

 

Key Lesson Marker

The Negative Exponent Law in general form is,

To represent a negative exponent as a positive exponent, rewrite its positive index accompanied by its base in the denominator position. Conversely, if the index is negative in the denominator, rewrite its positive index with its base in the numerator.