This post shows how I use a reciprocal pairs approach to introduce zero and negative integer exponents to my students. Several exercises that I have used to make students more comfortable using zero and negative integer exponents are also included. It seems to me that students learn to tolerate negative exponents, but they are not really comfortable using negative exponents.
I begin my intuitive discussion of negative exponents by creating a table of the first 4 positive integer powers of an integer such as 6 for example. Refer to the table below.
From this point, the discussion goes something like this:
- What do we do to 1,296 to get 216? What do we do to 216 to get 36?
- Some groups immediately notice that we divide by 6 to get the next number in the list, and other groups require a bit of coaching to see that we divide by 6 to get the next number.
- In order to maintain the pattern, it’s clear that the fifth row of the table must be 60 = 1.
- To get the remaining rows, I say, “Reduce the exponent by one and divide by whole number six to get the next fraction in the list.” (Some students see a rational number like 3/8 only as a single entity, not one whole number divided by another whole number. Also some groups need to be reminded how a fraction is divided by a whole number.)
- As shown in the diagram, I bracket the reciprocal pairs of numbers in the list.
- Now the important question; “What is the relationship between bracketed pairs of numbers?”
- Some groups immediately notice the reciprocal pair relationship, and other groups need a little coaching to see the relationship. It takes a lot of coaching to get them to say out loud, “They are reciprocal pairs because all products of a pair of bracketed numbers equals 1.”
- I point out that we could do the same process with any nonzero real number, but the math might be a little messy.
Shown below is a reciprocals pairs table for -2/5 and -5/2. If a teacher decides to show his/her class a reciprocal pairs table for two fractions, and fraction operation skills are fragile, I strongly advise that the implied multiplication and division of fractions in the table be explicitly demonstrated. It’s sufficient to demonstrate multiplication and division of fractions for one or two lines in the table.
After exploring one or two tables of reciprocal number pairs, I give my students an informal summary of what can be deduced from the table. Of course, the concepts that I discuss depends on the class I’m teaching. Base b equals any nonzero real number, p is an integer, and x and y are real numbers. Students should use their calculators to verify the specific examples.
| Concept | Example |
|---|---|
| b0 = 1 | 3.140 = 1 |
| 1 / (x/y) = y/x | 1 / (-7/2) = -2/7 |
| All real numbers have an implied exponent of one. | π = π1 |
| Changing the sign of the exponent of a number gives us the reciprocal of the number. | 4-1 = 1/4 |
| If integer n > 0, bn means repeated multiplication of b by itself n times. | 43 = 4•4•4 = 64 |
| If integer n > 0, b-n means repeated multiplication of 1/b by itself n times. | 4-3 = (1/4)•(1/4)•(1/4) = 1/64 |
| b-p = (1/b)p | (1.2)-5 = (5/6)5 |
| bp = (1/b)-p | (-2.25)3 = (-1/2.25)-3 |
| xy = y / (1/x) | 0.008x = x/125 |
| If x > 0, Log(x) = -Log(1/x) because logarithms are exponents. | Log(5) = -Log(0.2) ≈ 0.699 |
| The graphs of y = a(bx) and y = a(1/b)-x are equal. | See graph below. |
| The graphs of y = a(bx) and y = a(b-x) are reflection images of each other over the y-axis because the graphs of y = f(x) and y = f(-x) are always mirror images over the y-axis. | See graph below. |
I will close this post by showing you some practice exercises that I have used to promote understanding of negative exponents, and to increase student comfort level in working with negative exponents. One may think that these exercises are nothing more than mental gymnastics, however, calculus students need to know how to rewrite expressions in terms of negative and fractional exponents. Readers can download my free handout, Properties of Exponents and Logarithms, by going to the algebra and pre-calculus tab in our instructional content page. This handout is filled with examples demonstrating the laws of exponents and logarithms.
