Multiplication Ignoring Powers (MIP)

Another law of exponents frequently misunderstood by students is

\begin{displaymath}(a^m)(b^m) = (ab)^m. \end{displaymath}

This means that such statements as
\begin{eqnarray*} 4x^2 & = & 2^2 x^2 = (2x)^2, \qquad\mbox{and} \\ 3\sqrt{x} & = & 9^{1/2}x^{1/2} = (9x)^{1/2} = \sqrt{9x}. \end{eqnarray*}

are correct. But many students ignore the significance of having identical powers in these multiplications. They make statements like the following, all of which are incorrect:
\begin{displaymath}2x^{1/2} = \sqrt{2x}, \mbox{\hspace{0.6in} \epsfig{0.4in}{mip.eps}} \end{displaymath}

\begin{displaymath}-(3x)^2 = 9x^2, \mbox{\hspace{0.6in} \epsfig{0.4in}{mip.eps}} \end{displaymath}

\begin{displaymath}3(x+1)^2 = (3x+3)^2, \mbox{\hspace{0.6in} \epsfig{0.4in}{mip.eps}} \end{displaymath}

\begin{displaymath}\frac{3}{2x^2} = 3(2x)^{-2}. \mbox{\hspace{0.6in} \epsfig{0.4in}{mip.eps}} \end{displaymath}


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

Thomas L. Scofield 2003-09-04