## Confusing Negative and Fractional Exponents (CNFE)

Students can make a variety of mistakes when it comes
to working with exponents. Two of the most common are
**Multiplying Exponents that should be Added (MEA)**,
and **Adding Exponents that should be Multiplied (AEM)**.
This section does not deal with either of these, but rather
with a problem that some students have applying
two basic rules about exponents, the ones concerning
reciprocals and roots. Specifically, these are

respectively, where the understanding is that a square
root (
) is to be taken as
(
).
The first of these says that
a *factor* of the denominator (see the discussion on
**CES**) raised to a power (be it positive or negative)
may be written as a factor to the oppositite power of
the numerator (i.e., a power becomes , a
power becomes ). The only change is to
the *sign* of the exponent. An example of a *valid*
application of this rule is

The second rule shows how to write a root as a power,
which can be especially helpful in calculus when a derivative
is desired. Things like

Some students seem to confuse these two rules. The main
errors seem to come from students trying to reciprocate
the wrong thing

or from students putting a minus in when none is required

Top Algebra Errors Made by Calculus Students
(full document)

Full List of Grading Codes

Thomas L. Scofield
2003-09-04