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

\begin{displaymath}\frac{1}{x^m} \;=\; x^{-m} \qquad\mbox{and}\qquad \sqrt[q]{x^p} \;=\; x^{p/q}, \end{displaymath}

respectively, where the understanding is that a square root ( $\sqrt{\phantom{x}}$) is to be taken as ( $\sqrt[2]{\phantom{x}}$).

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 $(-2)$ power becomes $(+2)$, a $(3/4)$ power becomes $(-3/4)$). The only change is to the sign of the exponent. An example of a valid application of this rule is

\begin{displaymath}\frac{3}{2x^3} \;=\; \frac{3}{2}x^{-3} \qquad\mbox{or} \qquad 3(2)^{-1}x^{-3}. \end{displaymath}

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

\begin{displaymath}\begin{array}{lll} \ds{\sqrt[3]{3x^2}} & \mbox{may be writt... ... as} & \ds{\frac{\sqrt{5}x^{1/2}}{(x-2)^{1/2}}}. \end{array} \end{displaymath}

Some students seem to confuse these two rules. The main errors seem to come from students trying to reciprocate the wrong thing
\begin{displaymath}\frac{1}{x^2} = x^{1/2}, \mbox{\hspace{0.6in} \epsfig{0.5in}{cnfe.eps}} \end{displaymath}

\begin{displaymath}\frac{2}{x^{1/2}} = 2x^2, \mbox{\hspace{0.6in} \epsfig{0.5in}{cnfe.eps}} \end{displaymath}

or from students putting a minus in when none is required
\begin{displaymath}\sqrt[3]{x^2} = x^{-2/3}, \mbox{\hspace{0.6in} \epsfig{0.5in}{cnfe.eps}} \end{displaymath}

\begin{displaymath}\frac{3}{\sqrt{2x-1}} \;=\; \frac{3}{(2x-1)^{-1/2}} \;=\; 3(2x-1)^{1/2}. \mbox{\hspace{0.6in} \epsfig{0.5in}{cnfe.eps}} \end{displaymath}


Top Algebra Errors Made by Calculus Students (full document)
Full List of Grading Codes

Thomas L. Scofield 2003-09-04